The Bidimensionality Theory
and Its Algorithmic Applications
by
MASSACHUSETTS INS
OF TECHNOLOGY
MohammadTaghi Hajiaghayi
B.S., Sharif University of Technology, 2000
M.S., University of Waterloo, 2001
Submitted to the Department of Mathematics
JUN 2 8 2005
LIBRARIE
in partial fulfillment of the requirements for the degree of
DOCTOR OF PHILOSOPHY
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
June 2005
) MohammadTaghi Hajiaghayi, 2005. All rights reserved.
The author hereby grants to MIT permission to reproduce and
distribute publicly paper and electronic copies of this thesis document
in whole or in part.
Author...............
Department of Mathematics
April
A
29, 2005
Certified by ....................
Erik D. Demaine
Associate Professor of Electrical Engineering and Computer Science
Thesisupervisor
Accepted by ..............
"""',.
.
.. .........................
odolfo Ruben
Rodolfo Ruben Rosales
Chairman, Applied Mathematics Committee
Accepted by
..........................
f,
Pavel I. Etingof
Chairman, Department Committee on Graduate Students
ARCHIVES
2
The Bidimensionality Theory
and Its Algorithmic Applications
by
MohammadTaghi Hajiaghayi
Submitted to the Department of Mathematics
on April 29, 2005, in partial fulfillment of the
requirements for the degree of
DOCTOR OF PHILOSOPHY
Abstract
Our newly developing theory of bidimensional graph problems provides general techniques
for designing efficient fixed-parameter algorithms and approximation algorithms for NP-
hard graph problems in broad classes of graphs. This theory applies to graph problems
that are bidimensional in the sense that (1) the solution value for the k x k grid graph (and
similar graphs) grows with k, typically as Q(k 2 ), and (2) the solution value goes down when
contracting edges and optionally when deleting edges. Examples of such problems include
feedback vertex set, vertex cover, minimum maximal matching, face cover, a series of vertex-
removal parameters, dominating set, edge dominating set, r-dominating set, connected
dominating set, connected edge dominating set, connected r-dominating set, and unweighted
TSP tour (a walk in the graph visiting all vertices). Bidimensional problems have many
structural properties; for example, any graph embeddable in a surface of bounded genus
has treewidth bounded above by the square root of the problem's solution value. These
properties lead to efficient-often subexponential-fixed-parameter
algorithms, as well as
polynomial-time approximation schemes, for many minor-closed graph classes. One type
of minor-closed graph class of particular relevance has bounded local treewidth, in the
sense that the treewidth of a graph is bounded above in terms of the diameter; indeed, we
show that such a bound is always at most linear. The bidimensionality theory unifies and
improves several previous results. The theory is based on algorithmic and combinatorial
extensions to parts of the Robertson-Seymour Graph Minor Theory, in particular initiating
a parallel theory of graph contractions. The foundation of this work is the topological theory
of drawings of graphs on surfaces and our results regarding the relation (the linearity) of
the size of the largest grid minor in terms of treewidth in bounded-genus graphs and more
generally in graphs excluding a fixed graph H as a minor.
In this thesis, we also develop the algorithmic theory of vertex separators, and its
relation to the embeddings of certain metric spaces. Unlike in the edge case, we show
that embeddings into L 1 (and even Euclidean embeddings) are insufficient, but that the
additional structure provided by many embedding theorems does suffice for our purposes.
We obtain an O( lo-gn) approximation for min-ratio vertex cuts in general graphs, based
on a new semidefinite relaxation of the problem, and a tight analysis of the integrality gap
which is shown to be O('Iogn).
We also prove various approximate max-flow/min-vertex3
cut theorems, which in particular give a constant-factor approximation for min-ratio vertex
cuts in any excluded-minor family of graphs. Previously, this was known only for planar
graphs, and for general excluded-minor families the best-known ratio was O(log n). These
results have a number of applications. We exhibit an O(Vlo/g) pseudo-approximation for
finding balanced vertex separators in general graphs. Furthermore, we obtain improved
approximation ratios for treewidth: In any graph of treewidth k, we show how to find a tree
decomposition of width at most O(kx/lg-k), whereas previous algorithms yielded O(k log k).
For graphs excluding a fixed graph as a minor, we give a constant-factor approximation
for the treewidth; this via the bidimensionality theory can be used to obtain the first
polynomial-time approximation schemes for problems like minimum feedback vertex set
and minimum connected dominating set in such graphs.
Thesis Supervisor: Erik D. Demaine
Title: Associate Professor of Electrical Engineering and Computer Science
4
Acknowledgments
First and foremost, I am deeply indebted to my thesis advisor, Professor Erik Demaine, and my academic advisor, Professor Tom Leighton, for being enthusiastic and
genius supervisors. Erik and I were more friends than advisor and advisee and we
spent countless hours together exchanging puzzles, research ideas, and philosophies.
Between us, we have explored lots of plausible ideas in algorithm design, many two
or more times. His insights (technical and otherwise) have been invaluable to me.
Erik also helped me a lot with his useful suggestions on the presentation of the results and the writing style. I am grateful to Tom for his perceptiveness, and his deep
insights over the years on my research. He provided me a great source of inspiration.
Both Erik and Tom have been extremely generous in giving me a great deal of their
valuable time and sharing with me their ideas and insights. This research would have
been impossible without their helps and encouragements.
I thank Professor Uriel Feige, Professor Fedor Fomin, Professor Naomi Nishimura,
Professor Prabhakar Ragde, Professor Paul Seymour, Professor Dan Spielman, Professor Dimitrios Thilikos, and James Lee for fruitful collaborations and discussions
in the areas related to this thesis. I especially thank Paul Seymour for many helpful
discussions and for providing a portal into the Graph Minor Theory and revealing
some of its hidden structure that we use in this thesis. Thanks go to Professor Daniel
Kleitman who served on my committee, read my thesis, and gave me useful comments.
Furthermore, I would like to thank the staff and the faculty of the Department of
Mathematics and Computer Science and Artificial Intelligence Laboratory at MIT,
especially Kathleen Dickey and Linda Okun, for providing such a nice academic environment. I am also grateful to the researchers at IBM T.J. Watson Research Center
and Microsoft Research for two great summers in the research industry.
My friends here at MIT, namely Reza Alam, Mihai Badoiu, Saeed Bagheri,
Mohsen Bahramgiri, Eaman Eftekhary, Nick Harvey, Fardad Hashemi, Susan Hohenberger, Nicole Immorlica, Ali Khakifirooz, Bobby Kleinberg, Mahnaz Maddah,
Mahammad Mahdian, Vahab Mirrokni, Eddie Nikolova, Hazhir Rahmandad, Mohsen
5
Razavi, Navid Sabbaghi, Saeed Saremi, Anastasios Sidiropoulos, Ali Tabaei, Mana
Taghdiri, David Woodruff, Sergey Yekhanin, and many others played important roles
in making my life more enjoyable. Also, I would like to extend my appreciation
to thank my previous advisors in University of Waterloo, namely Professor Naomi
Nishimura, and in Sharif University of Technology, namely Professor Ebad Mahmoodian and Professor Mohammad Ghodsi, and my many other friends in Iran, Canada
and USA for their warmest friendship. I am blessed to have such excellent companies.
Last but not least, my dear parents and my family receive my heartfelt gratitude
for their sweetest support and never-ending love. I always feel the warmth of their
love, even now that we are so far away. I wish to thank my brother, Mahdi, and
my sisters, Monir and Mehri, for being the greatest source of love. This thesis is
dedicated to my parents and my family.
6
Contents
1 Introduction and Overview
13
1.1 Graph Terminology.
14
1.2 Graph Classes .
15
1.2.1
Definitions of Graph Classes ...........
15
1.3 Structural Properties.
17
1.3.1
Background.
18
1.3.2
Structure of Single-Crossing-Minor-Free Graphs
20
1.3.3
Structure of H-Minor-Free Graphs
20
1.3.4
Structure of Apex-Minor-Free Graphs ......
22
1.3.5
Grid Minors.
23
.......
1.4 Bidimensional Parameters/Problems.
24
1.5
Parameter-Treewidth Bounds.
25
1.6
Separator Theorems.
26
1.7
Local Treewidth .
27
1.8
Subexponential Fixed-Parameter Algorithms ......
29
1.9
Fixed-Parameter Algorithms for General Graphs ....
30
1.10 Polynomial-Time Approximation Schemes .......
32
1.11 Half-Integral versus Fractional Multicommodity Flow
34
1.12 Thesis Structure .
34
2 Approximation Algorithms for Single-Crossing-Minor-Free Graphs 37
2.1
Background.
38
2.1.1
38
Preliminaries.
7
2.1.2
2.2
2.3
Locally Bounded Treewidth
...................
39
Clique-sum Decompositions.
40
2.2.1
Relating Clique Sums to Treewidth and Local Treewidth . . .
40
2.2.2
Decomposition Algorithm.
41
Locally Bounded Treewidth of Single-Crossing-Minor-Free Graphs . .
47
2.3.1
Bounded Local Treewidth.
47
2.3.2
Local Treewidth and Layer Decompositions
..........
2.4 Approximating Treewidth.
2.5
2.6
50
54
Polynomial-time Approximation Schemes ................
58
2.5.1
General Schemes for Approximation on Special Classes of Graphs 58
2.5.2
Approximation Schemes for Single-Crossing-Minor-Free Graphs
Concluding Remarks.
59
63
3 Exponential Speedup of Fixed-Parameter Algorithms for Single-CrossingMinor-Free Graphs
65
3.1 Background ................................
67
3.1.1
Preliminaries
67
...........................
3.2 Fixed-Parameter Algorithms for Dominating Set
.
...........
70
3.3 Algorithms for Parameters Bounded by the Dominating-Set Number .
72
3.3.1
Variants of the Dominating Set Problem ............
74
3.3.2
Vertex Cover.
75
3.3.3
Edge Dominating Set .......................
76
3.3.4
Clique-Transversal Set.
77
3.3.5
Maximal Matching.
77
3.3.6
Kernels in Digraphs ........................
78
3.4 Fixed-Parameter Algorithms for Vertex-Removal Problems .
3.4.1
Feedback Vertex Set .
81
3.4.2
Improving Bounds for Vertex Cover ...............
82
84
3.5 Further Extensions.
3.6
79
86
Concluding Remarks.
8
4
Fixed-Parameter Algorithms for the (k, r)-Center in Planar Graphs
and Map Graphs
5
89
4.1
Preliminary Results ...........................
4.2
Combinatorial Bounds
4.3
(k, r)-Centers in Graphs of Bounded Branchwidth .........
4.4
Algorithms
4.5
Concluding Remarks ..........................
..
92
..........................
for the (k, r)-Center
Problem
94
97
. . . . . . . . . . . . . . . .
104
105
Subexponential Parameterized Algorithms on Bounded-Genus Graphs
and H-Minor-Free Graphs
5.1
5.2
5.3
5.4
Graphs on Surfaces
109
. . . . . . . . . . . . . . . .
5.1.1
Preliminaries.
5.1.2
Bounding the Representativity
. . . . . .
. . . .
.
. . . . . . . .
111
. . . . . . . . . . . .111
. . . . . . .
..
.
113
Bidimensional Parameters and Bounded-Genus Graphs ........
115
5.2.1
Definitions
.
116
5.2.2
Examples
.............................
5.2.3
Subexponential Algorithms and Planar Graphs
5.2.4
Parameter-Treewidth Bound for Bounded-Genus Graphs
. . .
119
5.2.5
Combinatorial Results and Further Improvements ......
.
124
5.2.6
Algorithmic
. . . . . . . . . . . .
Consequences
.
. . . . . . . . .
116
.
.......
. . . . .
118
.
125
H-Minor-Free Graphs ...........................
126
5.3.1
Almost-Embeddable Graphs and r-Dominating Set .....
5.3.2
H-Minor-Free Graphs and Dominating Set ...........
Concluding Remarks .
.
127
128
........................
133
6 Diameter and Treewidth in Minor-Closed Graph Families
137
7 Bidimensional Parameters and Local Treewidth
143
7.1 Definitions and Preliminary Results ......
7.2 Combinatorial Lemmas .........................
7.3
Main Theorem
..............................
9
.
......
144
.
146
148
7.4 Algorithmic Consequences and Concluding Remarks ..........
151
8 Graphs Excluding a Fixed Minor have Grids Almost as Large as
Treewidth
155
8.1 Overview of Proof of Main Theorem ...................
157
8.2 Proof of Main Theorem.
158
.........................
9 Improved Approximation Algorithms for Minimum-Weight Vertex
Separators and Treewidth
165
9.1 Related Work.
.......
168
..
9.2 Results and Techniques ................
.......
169
..
9.3 A Vector Program for Minimum-Ratio Vertex Cuts
.......
172
..
.......
173
..
9.4
9.3.1
The Quadratic Program
...........
9.3.2
The Vector Relaxation ............
.......
175
...
9.3.3
Adding Valid Constraints.
.......
175
..
Algorithmic Framework for Rounding ........
.......
9.4.1
Line Embeddings and Vertex Separators . .
.......
179
..
9.4.2
Line Embeddings and Distortion
.......
181
...
9.4.3
Analysis of the Vector Program
.......
182
..
......
.......
.177
Approximate Max-Flow/Min-Vertex-Cut Theorems
.......
9.5.1
Rounding to Vertex Separators
.......
186
..
9.5.2
The Rounding .................
.......
186
..
9.5.3
Excluded Minor Families ...........
.......
188
..
9.6
An Integrality Gap for the Vector Program .....
.......
188
..
9.7
Balanced Vertex Separators and Applications
.......
191
..
9.5
.......
. . .
......................
.184
.191
9.7.1
More General Weights
9.7.2
Reduction from Min-Ratio Cuts to Balanced Separators.
193
9.7.3
Approximating Treewidth .
194
10 Open Problems Regarding Bidimensionality
10
199
List of Figures
1-1 Interesting classes of graphs. Arrows point from more specific classes
to more inclusive classes ..........................
16
1-2 Example of a 5-sum of two graphs.
...................
19
2-1 Examples of single-crossing graphs
...................
39
2-2 The graph V8............................................
.
49
2-3 The replacement of the part of path P between a and b by edge {a, b}.
51
2-4 A graph and two branch decompositions of it. The first has width 4
and the second has width 3. .......................
4-1
A partially triangulated
5-1 Splitting a noose
(12 x 12)-grid.
55
. . . . . . . . . . . . .....
106
..............................
122
6-1 Construction of the minor k x k grid K
.................
140
6-2 In the k x k grid K, we (a) lift the vertex v', (b) contract the adjacent
columns, and (c) contract the adjacent rows, to form a (k- 2) x (k-2)
grid K'. Vertex v' is adjacent to all vertices in the grid, though the
figure just shows four neighbors for visibility ...............
140
7-1 An augmented 12 x 12 grid with span 8. ................
7-2 Left: The grid H, the points in S", and their grouping. Here
Right: Construction of the minor
points
in...
.........
x
...
11
146
= 6.
grid R passing through the
...............
. ... 147
12
Chapter
1
Introduction and Overview
The newly developing theory of bidimensional graph problems, developed in a series of
papers [74, 72, 64, 65, 63, 66, 62, 73, 71, 70], provides general techniques for designing
efficient fixed-parameter algorithms and approximation algorithms for NP-hard graph
problems in broad classes of graphs. This theory applies to graph problems that are
bidimensional in the sense that (1) the solution value for the k x k grid graph (and
similar graphs) grows with k, typically as Q(k2 ), and (2) the solution value goes
down when contracting edges and optionally when deleting edges. Examples of such
problems include feedback vertex set, vertex cover, minimum maximal matching, face
cover, a series of vertex-removal parameters, dominating set, edge dominating set, Rdominating set, connected dominating set, connected edge dominating set, connected
R-dominating set, and unweighted TSP tour (a walk in the graph visiting all vertices).
Bidimensional problems have many structural properties; for example, any graph
in an appropriate minor-closed class has treewidth bounded above in terms of the
problem's solution value, typically by the square root of that value. These properties lead to efficient-often
subexponential-fixed-parameter
algorithms, as well as
polynomial-time approximation schemes, for many minor-closed graph classes. One
type of minor-closed graph class of particular relevance has bounded local treewidth,
in the sense that the treewidth of a graph is bounded above in terms of the diameter;
indeed, such a bound is always at most linear.
The bidimensionality theory unifies and improves several previous results. The
13
theory is based on algorithmic and combinatorial extensions to parts of the RobertsonSeymour Graph Minor Theory, in particular initiating a parallel theory of graph
contractions. The foundation of this work is the topological theory of drawings of
graphs on surfaces.
This chapter is organized as follows. We start with our graph terminology in
Section 1.1. Section 1.2 defines the various graph classes of increasing generality
to which bidimensionality theory applies.
Section 1.3 describes several structural
properties of graphs in these classes, in particular from Graph Minor Theory, that
form the basis of bidimensionality. Section 1.4 defines bidimensional parameters and
problems and gives some examples. Section 1.5 describes one of the main structural
properties of bidimensionality, namely, that the treewidth is bounded in terms of the
parameter value. Sections 1.6-1.11 describe several consequences of bidimensionality
theory: separator theorems, bounds on local treewidth, fixed-parameter algorithms,
and polynomial-time approximation schemes. Finally in Section 1.12, we describe the
structure of this thesis.
1.1
Graph Terminology
We assume the reader is familiar with general concepts of graph theory such as graphs,
trees, and planar graphs. The reader is referred to standard references for appropriate
background [40]. For exact definitions of various NP-hard problems in this paper, the
reader is referred to Garey and Johnson's seminal book [99]. Here we review a few
terms used in this thesis; others will be defined in the context of their use.
Throughout this thesis, all graphs are finite, simple, and undirected, unless indicated otherwise. A graph G is represented by G = (V, E), where V (or V(G)) is
the set of vertices and E (or E(G)) is the set of edges; we use n to denote IVI when
G is clear from context. An edge e in a graph G between u and v is denoted by
{u, v} or, equivalently, {v, u}. Here, vertices u and v are called the endpoints of e.
A graph G' = (V', E') is a subgraph of G if V' C V and E' C E, and is an induced
subgraph of G, denoted by G[V'], if in addition E' contains all edges of E that have
14
both endpoints in V'. The (disjoint) union of two disjoint graphs G1 and G2 , G1 UG2 ,
is a graph G formed by merging vertex and edge sets, so that V(G) = V(G 1) UV(G 2)
and E(G) = E(G1 ) U E(G 2).
We define the r-neighborhood of a set S C V(G), denoted by Ne(S), to be the set
of vertices at distance at most r from at least one vertex of S C V(G); if r = 1, we
simply use the notation NG(S) and if S = {v}, we simply use the notation N~(v).
We also define NG[v] = NG(v) - {v}.
The diameter of G, denoted by diam(G), is
the maximum over all distances between pairs of vertices of G. An n-clique (Kn)
is an n-vertex graph in which every pair of vertices is connected by an edge. The
vertices of the graph Kn,m can be partitioned into sets V1 and V2 such that IV I= n,
IV21= m, and the edge set consists of all edges {u, v} such that u E V1 and v E V2 .
A graph G = (V, E) is k-connected if for any S C V(G) such that SI < k, G[V - S]
is connected.
1.2
Graph Classes
In this section, we introduce several families of graphs, each playing an important role
in both the Graph Minor Theory and the bidimensionality theory. Refer to Figure
1-1. All of these graph classes are generalizations of planar graphs, which are wellstudied in algorithmic graph theory. Unlike planar graphs and map graphs, every
other class of graphs we consider can include any particular graph G; of course, this
inclusion requires a bound or excluded minor large enough depending on G. This
property distinguishes this line of research from other work considering exclusion of
particular minors, e.g., K 3, 3, K5 , or K 6 .
1.2.1
Definitions of Graph Classes
The first three classes of graphs relate to embeddings on surfaces. A graph is planar
if it can be drawn in the plane (or the sphere) without crossings. A graph has genus
15
Figure 1-1: Interesting classes of graphs. Arrows point from more specific classes to
more inclusive classes.
at most g if it can be drawn in an orientable surface of genus g without crossings.l A
class of graphs has bounded genus if every graph in the class has genus at most g for
a fixed g.
Given an embedded planar graph and a two-coloring of its faces as either nations
or lakes, the associated map graph has a vertex for each nation and an edge between
two vertices corresponding to nations (faces) that share a vertex. The dual graph is
defined similarly, but with adjacency requiring a shared edge instead of just a shared
vertex. Map graphs were introduced by Chen, Grigni, and Papadimitriou [58] as a
generalization of planar graphs that can have arbitrarily large cliques. Thorup [162]
gave a polynomial-time algorithm for constructing the underlying embedded planar
graph and face two-coloring for a given map graph, or determining that the given
graph is not a map graph.
We view the class of map graphs as a special case of taking fixed powers of a
family of graphs. The kth power Gk of a graph G is the graph on the same vertex set
V(G) with edges connecting two vertices in Gk precisely if the distance between these
vertices in G is at most k. For a bipartite graph G with bipartition V(G) = U U W,
the half-square G2 [U] is the graph on one side U of the partition, with two vertices
adjacent in G 2 [U] precisely if the distance between these vertices in G is 2. A graph is
1
This definition also includes graphs that can be drawn in non-orientable surfaces of low genus,
because if a graph has non-orientable genus g, then it has orientable genus at most 2g.
16
a map graph if and only if it is the half-square of some planar bipartite graph [58]. In
fact, this translation between map graphs and half-squares is constructive and takes
polynomial time.
The next three classes of graphs relate to excluding minors. Given an edge e =
{v, w} in a graph G, the contraction of e in G is the result of identifying vertices v and
w in G and removing all loops and duplicate edges (the resulting graph is denoted
by G/e). A graph H obtained by a sequence of such edge contractions starting from
G is said to be a contraction of G. A graph H is a minor of G if H is a subgraph of
some contraction of G. We use the notation H < G (resp. H _c G) for H is a minor
(a contraction) of G. A graph class C is minor-closed if any minor of any graph in C
is also a member of C. A minor-closed graph class C is H-minor-free if H , C. More
generally, we use the term "H-minor-free" to refer to any minor-closed graph class
that excludes some fixed graph H.
A single-crossing graph is a minor of a graph that can be drawn in the plane with
at most one pair of edges crossing. Note that a single-crossing graph may not itself
be drawable with at most one crossing pair of edges; see Section 2.1. Such graphs
were first defined by Robertson and Seymour [142]. A minor-closed graph class is
single-crossing-minor-free if it excludes a fixed single-crossing graph.
An apex graph is a graph in which the removal of some vertex leaves a planar
graph. A graph class is apex-minor-free if it excludes some fixed apex graph. Such
graph classes were first considered by Eppstein [85, 87], in connection to the notion
of bounded local treewidth as described in Section 1.7.
The next section describes strong structural properties of the last three classes of
graphs (minor-excluding classes) in terms of the first two classes of graphs (embeddable on surfaces) and other ingredients.
1.3 StructuralProperties
Graphs from single-crossing-minor-free and H-minor-free graph classes have powerful
structural properties from the Graph Minor Theory. First we need to define treewidth,
17
pathwidth, and clique sums.
1.3.1
Background
Treewidth:
Many difficult graph problems can be solved efficiently when the input
is restricted to graphs of bounded treewidth (see e.g., Bodlaender's survey [31]). The
notion of treewidth first was introduced by Robertson and Seymour [143]. To define
this notion, first we consider a representation of a graph as a tree, called a tree
decomposition. Precisely, a tree decomposition of a graph G = (V, E) is a pair (T, X)
in which T = (I, F) is a tree and X = {XiI i E I} is a family of subsets of V(G) such
that
1. UiE Xi = v;
2. for each edge e = {u, v} E E, there exists an i E I such that both u and v
belong to Xi; and
3. for all v E V, the set of nodes {i E I I v E Xi} forms a connected subtree of T.
To distinguish between vertices of the original graph G and vertices of T in the tree
decomposition, we call vertices of T nodes and their corresponding Xi's bags. The
width of the tree decomposition is the maximum size of a bag in X minus 1. The
treewidth of a graph G, denoted by tw(G), is the minimum width over all possible
tree decompositions of G. A tree decomposition is called a path decomposition if
T = (I, F) is a path. The pathwidth of a graph G, denoted pw(G), is the minimum
width over all possible path decompositions of G.
A graph of bounded treewidth is a graph of treewidth at most k, for k a constant
independent of the size of the graph. A related notion is that of a k-tree [152], a
graph G such that either G is a k-clique or G has a vertex u of degree k such that
u is adjacent to a k-clique, and the graph obtained by deleting u and all its incident
edges is a k-tree. It has been shown that for any k, the class of graphs of treewidth at
most k is equivalent to the class of partial k-trees, that is, subgraphs of k-trees [127].
18
h
G =G
1
G2
Figure 1-2: Example of a 5-sum of two graphs.
Clique sum:
The notion of clique sums goes back to characterizations of K 3 ,3-
minor-free and K5 -minor-free graphs by Wagner [164] and serves as an important
tool in the Graph Minor Theory. Suppose G1 and G2 are graphs with disjoint vertex
sets and let k > 0 be an integer. For i = 1, 2, let Wi C V(Gi) form a clique of size k
and let G' be obtained from Gi by deleting some (possibly no) edges from the induced
subgraph Gi[Wi] with both endpoints in Wi. Consider a bijection h: W1 -+ W2. We
define a k-sum G of G 1 and G2, denoted by G = G1 EDkG2 or simply by G = G1
G2,
to be the graph obtained from the union of G' and G' by identifying w with h(w) for
all w E W1 . The images of the vertices of W1 and W 2 in G1 Ok G2 form the join set.
Note that each vertex v of G has a corresponding vertex in G1 or G2 or both. It
is also worth mentioning that
is not a well-defined operator: it can have a set of
possible results. More specifically, the result of ® will depend on which (if any) edges
are removed from the cliques as well as which bijection is selected, so the operation
o can have a set of possible
results, and hence is not well-defined. A series of k-sums
(not necessarily unique) that generate a graph G is called a decomposition of G into
clique-sum operations.
Figure 1-2 demonstrates an example of a 5-sum operation.
19
The reader is referred to [77] to see more results on clique-sum classes.
1.3.2
Structure of Single-Crossing-Minor-Free Graphs
The structure of single-crossing-minor-free graphs can be described as follows:
Theorem
1.1 ([142]). For any fixed single-crossing graph H, every H-minor-free
graph can be obtained by a sequence of k-sums, 0 < k < 3, of planar graphs and
graphs of bounded treewidth, where the bound on treewidth depends on H.
This theorem generalizes characterizations of K 3,3-minor-free and K 5-minor-free
graphs [164]. A graph is K3 ,3-minor-free if and only if it can be obtained by k-sums,
0 < k < 2, of planar graphs and K 5 . A graph is K 5-minor-free if and only if it can
be obtained by k-sums, 0
k < 3, of planar graphs and V8 (the length-8 cycle C8
together with eight edges joining diametrically opposite vertices).
This structural property of single-crossing-minor-free graphs has since been strengthened to ensure that the summands are minors of the original graph and to provide
algorithms for finding the decomposition:
Theorem 1.2 ([72], see also Chapter 2). For any fixed single-crossing graph H, there
is an O(n 4 )-time algorithm to compute, given an H-minor-free graph G, a decomposition of G as a sequence of k-sums, 0
k < 3, of planar graphs and graphs of
bounded treewidth (where the bound on treewidth depends on H), each of which is a
minor of G.
1.3.3
Structure of H-Minor-Free Graphs
The structure of H-minor-free graphs is described by a deep theorem of Robertson
and Seymour [149]. Intuitively, their theorem says that, for every graph H, every
H-minor-free graph can be expressed as a "tree structure" of pieces, where each piece
is a graph that can be drawn in a surface in which H cannot be drawn, except
for a bounded number of "apex" vertices and a bounded number of "local areas of
non-planarity" called vortices. Here the bounds depend only on H.
20
Roughly speaking, we say that a graph G is h-almost embeddablein a surface S if
there exists a set X of size at most h of vertices, called apex vertices or apices, such
that G - X can be obtained from a graph Go embedded in S by attaching at most h
graphs of pathwidth at most h to Go within h faces in an orderly way. More precisely,
a graph G is h-almost embeddable in S if there exists a vertex set X of size at most
h (the apices) such that G - X can be written as Go U G 1 U .. U Gh, where
1. Go has an embedding in S;
2. the graphs Gi, called vortices, are pairwise disjoint;
3. there are faces F1 ,..., F of Go in S, and there are pairwise disjoint disks
D1,. . ., Dh in S, such that for i = 1,..
h, Di C Fi and Ui := V(Go)n V(Gi) =
V(Go) n Di; and
4. the graph Gi has a path decomposition (Bu)UEU,of width less than h, such that
u E Bu for all u E Ui. The sets Bu are ordered by the ordering of their indices
u as points along the boundary cycle of face Fi in Go.
An h-almost embeddable graph is apex-free if the set X of apices is empty.
Now, the deep result of Robertson and Seymour is as follows:
Theorem
1.3 ([149]). For every graph H, there exists an integer h > 0 depending
only on V(H)I such that every H-minor-free graph can be obtained by at most h-
sums of graphs that are h-almost-embeddablein some surfaces in whichH cannot be
embedded.
In particular, if H is fixed, any surface in which H cannot be embedded has
bounded genus. Thus, the summands in the theorem are h-almost-embeddable in
bounded-genus surfaces.
Another way to view Theorem 1.3 is that every H-minor-free graph G has a
tree decomposition (T, X) such that, for each node i E V(T), the induced subgraph
G[Xi] augmented with additional edges to form a clique on the vertices that overlap
with the parent's bag, and a clique on the vertices that overlap with each child's
21
bag, is h-almost-embeddable in a bounded-genus surface. (This augmented graph is
called the torso [Xi] in, e.g., [103, 81].) The intersections between bag Xi and its
parent's bag, and with each child's bag, correspond to the join sets in the clique-sum
decomposition. Our development primarily follows the original clique-sum viewpoint
of Robertson and Seymour, but we will also occasionally view the sums as being
organized into the tree T.
Theorem 1.3 is very general and appeared in print only recently. However already
several nice applications
(see e.g. [39, 103, 75]) are known.
As observed by Seymour [154], the constructive proof of Theorem 1.3 in [149]also
establishes the following algorithmic result. (Grohe [103] also shows how to obtain a
similar result using Robertson and Seymour's theorem that every minor-closed class
of graphs has a polynomial-time membership test [149].)
Theorem 1.4. [149, 154] For any graph H, there is an algorithm with running time
n ° (1 ) that either computes a clique-sum decomposition as in Theorem 1.3 for any
given H-minor-free graph G, or outputs that G is not H-minor-free. The exponent
in the running time depends on H.
1.3.4
Structure of Apex-Minor-Free Graphs
Apex-minor-free graph classes are an important subfamily of H-minor-free graph
classes. The general structural theorem for H-minor-free graphs applies in this context as well. However, reductions developed in [66] suggest that the decomposition
can be restricted to a particular form in the apex-minor-free case:
Conjecture 1.5 ([66]). For every graph H, there is an integer h > 0 depending only
on IV(H)I such that every H-minor-free graph can be obtained by at most h-sums of
graphs that are h-almost-embeddable in some surfaces in which H cannot be embedded
and whose apices are connected via edges only to vertices within vortices.
22
1.3.5
Grid Minors
The r x r grid is the canonical planar graph of treewidth O(r). In particular, an
important result of Robertson, Seymour, and Thomas [151]is that every planar graph
of treewidth w has an Q(w) x Q(w) grid graph as a minor. Thus every planar graph
of large treewidth has a grid minor certifying that its treewidth is almost as large (up
to constant factors). Recently, this result has been generalized to any H-minor-free
graph class:
Theorem 1.6 ([71], see also Chapter 8). For any fixed graph H, every H-minor-free
graph of treewidth w has an Q(w) x Q(w) grid as a minor.
Thus the r x r grid is the canonical H-minor-free graph of treewidth O(r) for any
fixed graph H. This result is also best possible up to constant factors. Chapter 8
discusses the remaining issue of bounding the constant factor and its dependence
on H.
A similar but weaker bound plays an important role in the Graph Minor Theory
[144]: for any fixed graph H and integer r > 0, there is an integer w > 0 such that
every H-minor-free graph with treewidth at least w has an r x r grid graph as a minor.
This result has been re-proved by Robertson, Seymour, and Thomas [151], Reed [141],
and Diestel, Jensen, Gorbunov, and Thomassen [80]. Among these proofs, the best
known bound on w in terms of r is that every H-minor-free graph of treewidth larger
than
3
r
2 0 51V(H)1
has an r x r grid as a minor [151]. Theorem 1.6 therefore offers an
exponential (and best possible) improvement over previous results.
Theorem 1.6 cannot be generalized to arbitrary graphs: Robertson, Seymour, and
Thomas [151]proved that some graphs have treewidth Q(r2 lg r) but have grid minors
only of size O(r) x O(r). The best known relation for general graphs is that having
treewidth more than 202r5 implies the existence of an r x r grid minor [151]. The best
possible bound is believed to be closer to
(r 2 Ig r) than
2
(r5), perhaps even equal
to O(r 2 g r) [151]. In fact, we [69] conjecture that the correct bound is O(r3 ).
23
1.4 Bidimensional Parameters/Problems
Bidimensionality has been introduced and developed in a series of papers [74, 72, 64,
65, 63, 66, 62, 73, 71, 70]. Although implicitly hinted at in [74, 72, 64, 65], the first
use of the term "bidimensional"
was in [63].
First we define "parameters" as an alternative view on optimization problems. A
graph parameter P (or just a parameter P, when it clear in the context) is a function
mapping graphs to nonnegative integers. The decision problem associated with P
asks, for a given graph G and nonnegative integer k, whether P(G)
k. Many
optimization problems can be phrased as such a decision problem about a graph
parameterP.
Now we can define bidimensionality.
A parameter
is g(r)-bidimensional
(or just
bidimensional) if it is at least g(r) in an r x r "grid-like graph" and if the parameter does not increase when taking either minors (g(r)-minor-bidimensional) or contractions (g(r)-contraction-bidimensional).
The exact definition of "grid-like graph"
depends on the class of graphs allowed and whether we are considering minor- or
contraction-bidimensionality.
For minor-bidimensionality and for any H-minor-free
graph class, the notion of a "grid-like graph" is defined to be the r x r grid, i.e., the
planar graph with r2 vertices arranged on a square grid and with edges connecting
horizontally and vertically adjacent vertices. For contraction-bidimensionality, the
notion of a "grid-like graph" is as follows:
1. For planar graphs and single-crossing-minor-free graphs, a "grid-like graph" is
an r x r grid partially triangulated by additional edges that preserve planarity.
2. For bounded-genus graphs, a "grid-like graph" is such a partially triangulated
r x r grid with up to genus(G) additional edges ("handles").
3. For apex-minor-free graphs, a "grid-like graph" is an r x r grid augmented with
additional edges such that each vertex is incident to 0(1) edges to nonboundary
vertices of the grid. (Here 0(1) depends on the excluded apex graph.)
Contraction-bidimensionality is so far undefined for H-minor-free graphs (or general
24
graphs).2
Examples of bidimensional parameters include the number of vertices, the diameter, and the size of various structures such as feedback vertex set, vertex cover,
minimum maximal matching, face cover, a series of vertex-removal parameters, dominating set, edge dominating set, R-dominating set, connected dominating set, connected edge dominating set, connected R-dominating set, and unweighted TSP tour
(a walk in the graph visiting all vertices).
(See [63, 62] for arguments of either
contraction- or minor-bidimensionality for the above parameters.) We also say that
the corresponding optimization problems based on these parameters, e.g., finding the
minimum-size dominating set, are bidimensional. With the exception of diameter, all
of these bidimensional problems are O(r 2)-bidimensional, which is the most common
case (and in some papers used as the definition of bidimensionality). Diameter is the
main exception, being only O(r)-contraction-bidimensional
for planar graphs, single-
crossing-minor-free graphs, and bounded-genus graphs, and only
(lg r)-contraction-
bidimensional for apex-minor-free graphs.
1.5 Parameter-Treewidth Bounds
The genesis of bidimensionality was in fact the notion of a parameter-treewidth bound.
A parameter-treewidth bound is an upper bound f(P(G)) on the treewidth of a graph
with parameter P. Given a graph parameter P, we say that a graph family F has
the parameter-treewidthpropertyfor P if there is a strictly increasing function f such
that every graph G E F has treewidth at most f(P(G)).
even be shown to be sublinear in k, often O(vi),
Parameter-treewidth
In many cases, f(k) can
where k = P(G) for a graph G.
bounds have been established for many parameters and graph
classes; see, e.g., [2, 116, 93, 6, 50, 123, 107, 64, 72, 74, 62, 66, 63].
Essentially
all of these bounds can be obtained from the theory of bidimensional parameters.
Thus bidimensionality is the most powerful method so far for establishing parameter2
For the parameters to which we have applied bidimensionality, contraction-bidimensionality
does
not seem to extend beyond apex-minor-freegraphs, but perhaps a suitably extended definition could
be found in the context of different applications or a "theory of graph contractions".
25
treewidth bounds, encompassing all such previous results for H-minor-free graphs.
The central result in bidimensionality that generalizes these bounds is that every
bidimensional parameter has a parameter-treewidth bound, in its corresponding family of graphs as defined in Section 1.4. More precisely, we have the following result:
Theorem
1.7 ([71, 62], see also Chapters 8 and 7). If the parameter P is g(r)-
bidimensional,then for every graphG in the family associatedwith the parameterP,
tw(G) = O(g- 1(P(G))). In particular, if g(r) = O(r2), then the bound becomes
tw(G)=O(PG).
This theorem is based on the grid-minor bound from Theorem 1.6 and the proof of
a weaker parameter-treewidth bound, tw(G) = (g-1(p(G)))(9S-(P(G))), established
in [62] (see also Chapter 7). The stronger bound of tw(G) = O(g- 1 (P(G))) was
obtained first for planar graphs [64](see also Chapter 4), then single-crossing-minorfree graphs [74, 72](see also Chapters
2 and 3), then bounded-genus
graphs [63, 73]
(see also Chapter 5), and finally apex-minor-free graphs for contraction-bidimensional
parameters and H-minor-free graphs for minor-bidimensional parameters [71] (Theorem 1.7 above).
We can extend the definition of g(r)-minor-bidimensionality to general graphs
by again defining a "grid-like graph" to be the r x r grid. Still we can obtain a
parameter-treewidth bound [151, 68], but the bound is weaker: tw(G) = 2(9-1
(
))5
1.6 Separator Theorems
If we apply the parameter-treewidth bound of Theorem 1.7 to the parameter of the
number of vertices in the graph, which is minor-bidimensional with g(r) = r2 , then we
immediately obtain the following (known) bound on the treewidth of an H-minor-free
graph:
Theorem 1.8 ([9, Proposition 4.5], [103, Corollary 24], [71]). For any fixed graph H,
every H-minor-free graph G has treewidth O( I/V(G)).
26
A consequence of this result is that every vertex-weighted H-minor-free graph
G has a vertex separator of size O(v/V(G)I)
whose removal splits the graph into
two parts each with weight at most 2/3 of the original weight [9, Theorem 1.2].
This generalization of the classic planar separator theorem has many algorithmic
applications; see e.g. [9, 5]. Also, this result shows that the structural properties
of H-minor-free graphs given by Theorem 1.3 are powerful enough to conclude that
these graphs have small separators, which we expect from such a strong theorem.
Chapter 8 discusses the issue of how tight a lead constant can be obtained in such
a result.
1.7
Local Treewidth
Eppstein [87] introduced the diameter-treewidth property for a class of graphs, which
requires that the treewidth of a graph in the class is upper bounded by a function of
its diameter. He proved that a minor-closed graph family has the diameter-treewidth
property precisely if the graph family excludes some apex graph. In particular, he
proved that any graph in such a family with diameter D has treewidth at most 220().
(A simpler proof of this result was obtained in [65] (see also Chapter 6).)
If we apply the parameter-treewidth
bound of Theorem 1.7 to the diameter pa-
rameter, which is contraction-bidimensional with g(r) = O(lg r) [65], then we immediately obtain the following stronger diameter-treewidth bound for apex-minor-free
graphs:
Theorem 1.9 ([71], see also Chapter 8). For any fixed apex graph H, every H-minorfree graph of diameter D has treewidth 2 0(D) .
This theorem is not the best possible. In some sense it is necessarily limited because it still does not exploit the full structure of H-minor-free graphs from Theorem
1.3. The difficulty is that, in a grid-like graph, the 0(1) edges from a vertex to nonboundary vertices can accumulate to make the diameter small. However, it is possible
to show that, effectively, not too many vertices can have such edges. This fact comes
27
from the property that there are a bounded number of apices in the clique sum decomposition of Theorem 1.3, and in an apex-minor-free graph, each apex cannot have
more than a bounded number of edges to "distant" vertices. Based on this fact, a
complicated proof establishes the following even stronger diameter-treewidth bound
in apex-minor-free graphs:
Theorem
1.10 ([66]). For any fixed apex graph H, every H-minor-free graph of
diameter D has treewidthO(D).
This diameter-treewidth bound is the best possible up to constant factors. Thus
this theorem establishes that, in minor-closed graph families, having any diametertreewidth bound is equivalent to having a linear diameter-treewidth bound. As mentioned before, no minor-closed graph families beyond apex-minor-free graphs can have
any diameter-treewidth bound. Theorem 1.10 is therefore the ultimate characterization of diameter-treewidth bounds in minor-closed graph families (up to constant
factors).
The proof of Theorem 1.10 is the basis for Conjecture 1.5. In fact, Theorem 1.10
would not be hard to prove assuming Conjecture 1.5.
The diameter-treewidth property has been used extensively in a slightly modified
form called the bounded-local-treewidthproperty, which requires that the treewidth of
any connected subgraph of a graph in the class is upper bounded by a function of its
diameter. For minor-closed graph families, these two properties are identical. Graphs
of bounded local treewidth have many similar properties to both planar graphs and
graphs of bounded treewidth, two classes of graphs on which many problems are
substantially easier. In particular, Baker's approach for polynomial-time approximation schemes (PTASs) on planar graphs [23] applies to this setting. As a result,
PTASs are known for hereditary maximization problems such as maximum independent set, maximum triangle matching, maximum H-matching, and maximum tile
salvage; for minimization problems such as minimum vertex cover, minimum dominating set, minimum edge-dominating set; and for subgraph isomorphism for a fixed
pattern [72, 87, 110]. Graphs of bounded local treewidth also admit several efficient
28
fixed-parameter algorithms. In particular, Frick and Grohe [96] give a general framework for deciding any property expressible in first-order logic in graphs of bounded
local treewidth. Theorem 1.10 substantially improves the running time of these algorithms, in particular improving the running time of the PTASs from 2220(1/)n ° (1 ) to
2°(l/e)nO(l), where n is the number of vertices in the graph.
1.8 Subexponential Fixed-Parameter Algorithms
A fixed-parameter algorithm is an algorithm for computing a parameter P(G) of a
graph G whose running time is h(P(G)) no(M)for some function h. The exponent O(1)
must be independent of G; thus the exponentiality of the algorithm is bounded by
the parameter P(G), and the dependence on n is only polynomial. A typical function
h for many fixed-parameter algorithms is h(k) = 2 0(k). In the last six years, several
researchers have obtained exponential speedups in fixed-parameter algorithms in the
sense that the h function reduces exponentially, e.g., to 2 ° (`).
For example, the
first fixed-parameter algorithm for finding a dominating set of size k in planar graphs
[3] has running time 0(8kn); subsequently, a sequence of subexponential algorithms
and improvements have been obtained, starting with running time 0(46¥3n)
then 0(2274n)
[116], and finally O(215.'13Jvk +
n3
[2],
+ k 4 ) [93]. Other subexponential
algorithms for other domination and covering problems on planar graphs have also
been obtained
[2, 6, 50, 123, 107].
All subexponential fixed-parameter algorithms developed so far are based on showing a sublinear parameter-treewidth bound and then using an algorithm whose running time is singly exponential in treewidth and polynomial in problem size. As mentioned in Section 1.5, essentially all sublinear treewidth-parameter bounds proved so
far can be obtained through bidimensionality. Theorem 1.7 and the techniques of
[62] yield the following general result for designing subexponential fixed-parameter
algorithms:
Theorem 1.11 ([71, 62], see also Chapters 8 and 7). Consider a g(r)-bidimensional
parameter P that can be computed on a graph G in h(w) no() time given a tree
29
decomposition of G of width at most w. Then there is an algorithm computing P on
any graph G in P's corresponding graph class, with running time [h(O(g-l(k))) +
2°(9- (k))] n° (1) . In particular, if g(r)
=
E)(r2 ) and h(w) = 20(w2),then this running
time is subexponential in k.
In particular, this result gives subexponential fixed-parameter algorithms for many
bidimensional parameters, including feedback vertex set, vertex cover, minimum maximal matching, a series of vertex-removal parameters, dominating set, edge dominating set, R-dominating set, clique-transversal set, connected dominating set, connected
edge dominating set, connected R-dominating set, and unweighted TSP tour.
For minor-bidimensional parameters, these algorithms apply to all H-minor-free
graphs. The next section describes to what extent these algorithms can be extended
to general graphs.
For contraction-bidimensional parameters, these algorithms apply to apex-minorfree graphs. On the other hand, subexponential fixed-parameter algorithms can be
obtained for dominating set, which is contraction-bidimensional, on H-minor-free
graphs [63] (see also Chapter
5), map graphs [64] (see also Chapter
4), and fixed
powers of planar graphs (or even fixed powers of H-minor-free graphs) [64, 63]. These
algorithms are necessarily more complicated than those produced from Theorem 1.11,
because apex-minor-free graphs are precisely the minor-closed graph classes for which
domatinating set has a parameter-treewidth bound [62] (see Chapter 7). An intriguing
open question is whether these techniques can be extended to other contractionbidimensional problems than dominating set, for fixed powers of H-minor-free graphs
and/or other classes of graphs.
1.9 Fixed-Parameter Algorithms for General Graphs
As mentioned in Section 1.5, minor-bidimensionality can be defined for general graphs
as well. In this section we show how the bidimensionality theory in this case leads to
a general class of fixed-parameter algorithms.
A major result from the Graph Minor Theory (in particular [147, 150]) is that
30
every minor-closed graph property is characterized by a finite set of forbidden minors.
More precisely, for any property P on graphs such that a graph having property P
implies that all its minors have property P, there is a finite set {(H, H 2,... , Hh of
graphs such that a graph G has property P if and only if G does not have Hi as a
minor for all i = 1,2,...,
h. The algorithmic consequence of this result is that there
exists an O(n3 )-time algorithm to decide any fixed minor-closed graph property, by
finitely many calls to an O(n 3 )-time minor test [147]. This consequence has been
used to show the existence of polynomial-time algorithms for several graph problems,
some of which were not previously known to be decidable [91].
However, all of these algorithmic results (except the minor test) are nonconstructive: we are guaranteed that efficient algorithms exist, but are not told what they
are. The difficulty is that we know that a finite set of forbidden minors exists, but
lack "a means of identifying the elements of the set, the cardinality of the set, or even
the order of the largest graph in the set" [91]. Indeed, there is a mathematical sense
in which any proof of the finite-forbidden-minors theorem must be nonconstructive
[97].
We can apply these graph-minor results to prove the existence of algorithms to
compute parameters, provided the parameters never increase when taking a minor.
For any fixed parameter and any fixed k > 0, there is an O(n 3 )-time algorithm that
decides whether a graph has parameter value
k. Unfortunately, the existence of
these algorithms does not necessarily imply the existence of a single fixed-parameter
algorithm that works for all k > 0, because the algorithms for individual k (in particular the set of forbidden minors) might be uncomputable. We do not even know
an upper bound on the running time of these algorithms as a function of n and k,
because we do not know the dependence of the size of the forbidden minors on k.
In [68], fixed-parameter algorithms are constructed for nearly all parameters that
never increase when taking a minor, with explicit time bounds in terms of n and k.
Essentially, by assuming a few very common properties of the parameter, we obtain
the generalized form of minor-bidimensionality.
Theorem 1.12 ([68]). Consider a parameter P that is positive on some g x g grid,
31
never increases when taking minors, is at least the sum over the connected components
of a disconnected graph, and can be computed in h(w) n o (1 ) time given a width-w tree
decomposition of the graph. Then there is an algorithm that decides whether P is at
most k on a graph with n vertices in [22 °(g )5 + h(20(gV ) 5)] n0 () time.
As mentioned in [68], a conjecture of Robertson, Seymour, and Thomas [151]
would improve the running time to h(O(k lg k)) n ° ( ) , which is 20(klgk)n ° (1) for the
typical case of h(w) = 2 °( w). This conjectured time bound almost matches the fastest
known fixed-parameter algorithms for several parameters, e.g., feedback vertex set,
vertex cover, and a general family of vertex-removal problems [91].
1.10
Polynomial-Time Approximation Schemes
Recently, the bidimensionality theory has been extended to obtain polynomial-time
approximation schemes (PTASs) for essentially all bidimensional parameters, including those mentioned above [70]. These PTASs are based on techniques that generalize
and in some sense unify the two main previous approaches for designing PTASs in
planar graphs, namely, the Lipton-Tarjan separator approach [132] and the Baker
layerwise decomposition approach [23]. The PTASs apply to H-minor-free graphs
for minor-bidimensional parameters and to apex-minor-free graphs for contractionbidimensional parameters. To achieve this level of generality, [70] uses the sublinear
parameter-treewidth
bound of Theorem 1.7 as well as an 0(1)-approximation
algo-
rithm for treewidth in H-minor-free graphs [90] (see also Chapter 9).
Before we can state the general theorem for constructing PTASs, we need to
define a few straightforward required conditions, which are commonly satisfied by
most bidimensional problems. The theorem considers families of problems in which
we are given a graph and our goal is to find a minimum-size set of vertices and/or
edges satisfying a certain property. Such a problem naturally defines a parameter
and therefore the notion of bidimensionality. A minor-bidimensional problem has the
separation property if it satisfies the following three conditions:
32
1. If a graph G has k connected components G1,G2,
... ,
Gk,
then an optimal
solution for G is the union of optimal solutions for each connected component Gi.
2. There is a polynomial-time algorithm that, given any graph G, given any vertex
cut C whose removal disconnects G into connected components G, G2, ... , Gk,
and given an optimal solution Si to each connected component Gi of G - C,
computes a solution S for G such that the number of vertices and/or edges
in S within the induced subgraph G[C U UiEIV(Gi)] consisting of C and some
connected components of G - C is EiEI Sil + O(ICl) for any I C {1, 2,..., k}.
In particular, the total cost of S is at most opt(G - C) + O(ICl).
3. Given any graph G, given any vertex cut C, and given an optimal solution
opt to G, for any union G' of some subset of connected components of G - C,
loptn G'I = lopt(G')l + O(ICl).
For contraction-bidimensional problems, the exact requirements on the problem
are slightly different but similarly straightforward. The main distinction is that the
connected components are always considered together with the cut C. As a result,
the merging algorithm in Condition 2 must take as input a solution to a generalized
form of the problem that does not count the cost of including all vertices and edges
from the cut C. We refer to [70] for the exact definition of the separation property
in this case.
Theorem 1.13 ([70]). Consider a bidimensionalproblem satisfying the separation
property. Suppose that the problem can be solved on a graph G with n vertices in
f(n, tw(G)) time. Suppose also that the problem can be approximated within a factor
of a in g(n) time. For contraction-bidimensionalproblems, supposefurther that both
of these algorithms also apply to the generalized form of the problem. Then there is a
(1 + )-approximation algorithm whose running time is O(nf(n, O(a2 /e)) + n3 g(n))
for the corresponding graph class of the bidimensional problem.
This result shows a strong connection between subexponential fixed-parameter
tractability and approximation algorithms for combinatorial optimization problems
33
on H-minor-free graphs. In particular, this result yields a PTAS for the following
minor-bidimensional problems in H-minor-free graphs: feedback vertex set, face cover
(defined just for planar graphs), vertex cover, minimum maximal matching, and a
series of vertex-removal problems. Furthermore, the result yields a PTAS for the
following contraction-bidimensional problems in apex-minor-free graphs: dominating
set, edge dominating set, R-dominating set, connected dominating set, connected
edge dominating set, connected R-dominating set, and clique-transversal set.
1.11
Half-Integral versus Fractional Multicommodity Flow
Chekuri, Khanna, and Shephard [54]proved that, for planar graphs, the gap between
the optimal half-integral multicommodity flow and the optimal fractional multicommodity flow is at most a polylogarithmic factor. Also, they gave a combinatorial proof
of the result that, for planar graphs, the gap between the maximum flow and the minimum cut in product multicommodity flow (and thus uniform multicommodity flow)
instances is at most a constant factor. The latter result was proved before by Klein,
Plotkin, and Rao for H-minor-free graphs using primal-dual methods [120], and has
many applications in embeddings of H-minor-free graphs. As mentioned by Chekuri
et al. [54],our Theorem 1.6 can be used to generalize the half-integral/fractional gap
bound and the combinatorial proof of the max-flow/min-cut gap bound to H-minorfree graphs.
1.12 Thesis Structure
This thesis is organized as follows. We start by demonstrating structural properties
of single-crossing-minor-free graphs and their consequences in designing polynomialtime approximation algorithms and subexponential fixed-parameter algorithms for
a wide variety of graph problems in Chapters 2 and 3. In Chapter 4, we design
fixed-parameter algorithms for the (k, r)-center problem, a generalization of the dom34
inating set problem, on planar graphs and map graphs. At the end of this chapter,
we introduce the concept of bidimensionality for planar graphs. In Chapter 5, we
formally define the concept of bidimensionality for bounded-genus graphs. In addition, we introduce a general approach for developing algorithms on H-minor-free
graphs, based on the corresponding algorithms on bounded-genus graphs and structural results about H-minor-free graphs at the heart of Robertson and Seymour's
graph-minors work. Chapters 6 and 7 extend the concept of bidimensionality for
apex-minor-free graphs and H-minor-free graphs. In Chapter 8, we prove the linearity of the size of grid minors in terms of the treewidth, by which we improve several
combinatorial bounds and running times of algorithms in two previous Chapters 6
and 7. In Chapter 9, we develop the algorithmic theory of vertex separators and its
relation to the embeddings of certain metric spaces, by which we improve the approximation factor of treewidth for both general graphs and H-minor-free graphs. We also
mention how these improvements can be coupled with bidimensionality to obtain the
first polynomial-time approximation schemes for problems like minimum connected
dominating set and minimum feedback vertex set in apex-minor-free graphs and Hminor-free graphs. Finally, in Chapter 10, we demonstrate some major directions for
future research in the theory of bidimensionality.
It is worth mentioning that the journal versions of Chapters 2, 3, 4, 6, and 7
appeared in order in [72, 74, 64, 65, 62] and the conference versions of Chapters 5, 8, 9,
and this chapter appeared in order in [63, 71, 90, 67]. In particular, each chapter is
joint work with the set of collaborators listed in the corresponding references.
35
36
Chapter 2
Approximation Algorithms for
Single-Crossing-Minor-FreeGraphs
The development of algorithms for NP-complete problems on restricted classes of
graphs has resulted in structural characterizations of algorithmic utility. For example,
algorithms for graphs of bounded treewidth rely on techniques using separator properties resulting from tree decompositions. In this chapter we focus on graph classes obtained by excluding a single-crossing graph as a minor. We present a polynomial-time
algorithm that determines a clique-sum decomposition of such a graph, a representation of the graph as a collection of planar graphs and graphs of small treewidth Our
result generalizes previous decomposition results for graphs excluding special singlecrossing graphs such as K3 ,3 [20] and K5 [118]. A second structural property is that of
locally bounded treewidth which allows a layer decomposition, enabling us to represent
a graph as a collection of subgraphs, each of bounded treewidth (formal definitions
follow in Section 2.3). In order to take advantage of the bound on treewidth of the
subgraphs, we give a tree decomposition algorithm for this special case, adding to the
toolkit of tree decomposition algorithms for small bounds on treewidth [16, 133, 153]
and r-outerplanar graphs [2]that reduce the prohibitively high constant factors found
in algorithms
for more general graphs [125, 35, 32].
Included among the results in this chapter are several applications of our structural characterizations and algorithms.
Using clique-sum decompositions, we ob37
tain the first constant-factor approximation algorithm for treewidth of nonplanar
graphs. Furthermore, we use properties of layer decompositions to form polynomialtime approximation schemes for a range of maximization and minimization problems
(Section 2.5), as well as fixed-parameter algorithms for dominating set and related
problems (Section 3.5).
We present decomposition results followed by algorithms for NP-complete problems. Sections 2.1 through 2.3 present general results from which applications are
derived in Sections 2.4 and 2.5. First, in Section 2.1, we introduce the concepts used
throughout the chapter (though some of them are defined intuitively in Chapter 1).
Next, in Section 2.2, we demonstrate how graphs excluding single-crossing graphs
as minors can be characterized using a clique-sum decomposition.
The fact that
such graphs have bounded local treewidth is established in Section 2.3, which also
gives an algorithm for computing the tree decomposition of a local neighborhood in
a graph. The rest of the chapter contains results that follow from these properties:
an approximation algorithm for treewidth (Section 2.4), polynomial-time approximation schemes for optimization problems (Section 2.5), and summary of our results
and directions for further research (Section 2.6). We defer the applications regarding
fixed-parameter algorithms for dominating set and its variants to Chapter 3.
2.1
Background
2.1.1
Preliminaries
Recall that a graph is planar if it can be drawn in the plane so that its edges intersect
only at their endpoints; such a drawing is called an embedding. An embedding partitions the plane into connected regions called faces; the unbounded region is called
the outer face. A graph with all vertices on the outer face is called outerplanar, a
k-outerplanar graph has the property that k successive deletions of the vertices on
the outer face results in the empty graph.
We consider classes of graphs that are associated with single-crossing graphs, as
38
edge
XE
contraction
Figure 2-1: Examples of single-crossing graphs
defined by Robertson and Seymour [142]. As defined in Chapter 1, a single-crossing
graph is defined as a graph that is a minor of one that can be drawn in the plane with
at most one pair of edges crossing. Note that a single-crossing graph may itself not be
drawable in this fashion. Figure 2-1 shows to the left three examples of single-crossing
graphs; the third one cannot be drawn in the plane with only one crossing, but is
obtainable by edge contraction from the graph to the right, which can be so drawn.
The following lemma is a consequence of closure under minors; in contrast, the
class of graphs that can be drawn in the plane with at most one pair of edges crossing
is not closed under minors, as Figure 2-1 shows.
Lemma 2.1. If single-crossing-minor-free graph G excludes a single-crossing graph
H as a minor, any minor G' of G is also a single-crossing-minor-freegraph which
excludes H as a minor.
The following example demonstrates that single-crossing-minor-free graphs can be
considerably more complicated than single-crossing graphs. Form a graph on n = 6k
vertices by taking k copies of K3,3 and adding k - 1 edges to connect them into a
path-like structure.
This graph has
(n) crossings, but is K5-minor-free. In fact,
any graph can be shown to be a single-crossing-minor-free graph, where the excluded
single-crossing graph is a sufficiently large grid. However, our algorithms are really
only of interest when the excluded graph is small.
2.1.2
Locally Bounded Treewidth
The concept of treewidth can be generalized to that of locally bounded treewidth [87],
in which each local subgraph has treewidth bounded by a function of r.
39
Definition
2.2. The local treewidth of a graph G is the function ltwG
N: -- N
that associates with every r E N the maximum treewidth of an r-neighborhood in
G. We set ltwG(r) = maXvev(G){tw(G[NE(v)])}, and we say that a graph class C
has bounded local treewidth (or locally bounded treewidth) when there is a function
f N - N such that for all G E C and r E N, ltwG(r) < f(r). We say a graph class
C of bounded local treewidth has linear local treewidth, if f is linear in r. Finally,
for a function f: N -- N, we define the minor-closed class of graphs of bounded local
treewidthL(f) = {G: VH
-
G Vr > 0, ltwH(r) < f(r)}.
Well-known examples of minor-closed classes of graphs of bounded local treewidth
are graphs of bounded treewidth, planar graphs, and graphs of bounded genus [87].
Eppstein [87] showed that a minor-closed graph class C has bounded local treewidth
if and only if every graph in C is H-minor-free for some apex graph H (recall that in
any apex graph there exists a vertex whose deletion produces a planar graph). For a
simpler proof of this result, the reader might refer to Chapter 6.
The following lemma proves useful in our results:
Lemma 2.3. For any graph G and subgraph G' of G, ltwG' (k) < ltwG(k), for any
k > 0.
Proof. It is enough to observe that for any v E G' and k > 0, NG,(v) C NG(v). Thus
the removal of vertices of NG(v) \ NG,(v) from bags of a tree decomposition of NG(v)
results in a tree decomposition of NG,(v) with width at most that of G.
2.2
2.2.1
O
Clique-sum Decompositions
Relating Clique Sums to Treewidth and Local Treewidth
The following lemma shows how the treewidth changes when we apply a clique-sum
operation, which will play an important role in our approximation algorithms in
Section 2.4.
Lemma 2.4. For any two graphsG and H, tw(G ® H) < max{tw(G),tw(H)}.
40
Proof. We begin by observing that we can form a tree decomposition TD(G) of G
of width tw(G) and a tree decomposition TD(H) of H of width tw(H). For W the
set of vertices of G and H identified during the d operation, W is a clique in G and
in H. As vertices in a clique must appear together in a bag in any decomposition
of the graph [37], there exist a node a in TD(G) such that W C X and a node 3
in TD(H) such that W C Xp. Hence, we can form a tree decomposition of width
max{tw(G), tw(H)} of G G H by adding an edge between a in TD(G) and
in
TD(H).
[
To extend the result to graphs of bounded local treewidth in Lemma 2.9 (Section 2.3),
in Lemma 2.5 we establish treewidth properties for neighborhoods of graphs.
Lemma 2.5. For any graph G, any clique R of G, any v E R, and any k > 0,
tW(G[NG(R)])< tw(G[NG+l(v)]).
Proof. We note that all vertices in R-v are at distance 1 from v. Therefore NG1 (R) C
Nkl 1 (v), and the result follows from Lemma 2.3.
2.2.2
[]
Decomposition Algorithm
The main theorem of this section is a constructive version of the following nonalgorithmic result of Robertson and Seymour, itself used in our algorithm.
Theorem 2.6. [142]For any single-crossinggraph H, there is an integer
H
4
(depending only on H) such that every H-minor-free graph can be obtained by 0-, 1-,
2- or 3-sums of planar graphs and graphs of treewidth at most cH.
In the remainder of the chapter we will assume that CHis the smallest integer for which
Theorem 1 holds. Although previous algorithms have been developed for decomposing
specific single-crossing-minor-free graphs into series of clique sums (Asano [20]gave an
O(n)-time construction for K3 ,3-minor-free graphs and Kezdy and McGuinness [118]
gave an O(n 2 )-time construction for K5-minor-free graphs), ours is the first general
algorithm. For more examples of graph classes that can be characterized by cliquesum decompositions, see the work of Diestel [77, 78]. Theorem 2.8 below describes
41
a constructive algorithm to obtain a clique-sum decomposition of a single-crossingminor-free graph G that satisfies the additional property that the smaller graphs
are minors of G. The additional property is crucial for designing approximation
algorithms in Section 2.4.
To form a clique-sum decomposition of graphs that are minors of the original
graph, we consider graphs formed by first removing a set of vertices from the graph
and then reinserting the removed vertices in each resulting connected component.
More formally, we define a subset S C V to be a k-cut if the induced subgraph
G[V- S] is disconnected and IS}= k, and to be a strong k-cut if in addition G[V- S]
either has more than two connected components or each component has more than
one vertex. For S a strong cut that separates G into components G 1,.. ., Gh, we form
the augmented components induced by S, denoted Gi U K(S) for 1 < i < h, as the
graphs obtained from graphs G[V(Gi) U S] by adding an edge between each pair of
nonadjacent vertices in S. Each augmented component will contain as a subgraph a
clique on the vertices in S. The influence of strong cuts on augmented components
is fundamental in the proof of our theorem. By introducing a less strict definition of
strong cuts, we obtain a stronger version of an earlier lemma [118].
Lemma 2.7. Let S be a strong 3-cut of a 3-connectedgraphG = (V, E), and let
G1, G2 ,. .. , Gh denote the h components of G[V-S].
Then each augmented component
of G inducedby S, Gi U K(S), is a minor of G.
Proof. We first consider the case in which h > 3. By symmetry, it will suffice to show
that G1 U K(S) is a minor of G by forming the graph by a series of contractions in G.
Starting with G, we first contract all edges of G2 through Gh to obtain super-vertices
Y2through
Yh
(we use the term super-vertices to denote vertices that are the result
of the contraction of all the edges in specific connected subgraphs of G). Because G
is 3-connected, each super-vertex is adjacent to all vertices xl, x 2, and
3
in S. We
now contract the edge {x2, Y2} to form a super-vertex x and contract edges between
y3 through Yh and
x3
to form a super-vertex x 3. Again since each yi was adjacent to
each xj, we can conclude that x 1, x2, and x' form a clique and hence the resulting
42
graph is the augmented component G1 U K(S), as needed.
If instead h = 2, then by the definition of a strong cut G1 and G 2 each contain at
least two vertices. As in the previous case, it will suffice to show that G1 U K(S) is
a minor of G, as the argument for G2 U K(S) is symmetric. Again, we demonstrate
a series of contractions that results in G1 as well as a clique on the vertices Xl, 2,
and
X3
of S. We consider separately the cases in which G2 is a tree and G 2 contains
a cycle.
If G2 is a tree, then because G is 3-connected, there is a vertex yl in G2 that
neighbors xl, and similarly a vertex
y in G2 that neighbors
3
X3.
We contract
edges in G2 until there remain two super-vertices y and y, connected to xl and
X3
respectively. Since G2 is connected but acyclic, there will be a single edge between
y: and y. Moreover, since G is 3-connected, there is an edge between x2 and either
yI or y, say y. Finally, again because G is 3-connected, y must be adjacent to a
vertex of S other than
X3,
that is, either x2 or xl. In either case we can form a clique
on the vertices in S, by contracting edges {Xl, y'} and
or by contracting the edges {x2, y'} and
{X3,
{X3,
y'} if y' is adjacent to x2
y)} if y' is adjacent to xl.
Finally suppose that G2 has a cycle C. We claim that there are three vertexdisjoint paths connecting three vertices of C to three vertices of S in G2. By contracting these paths and then contracting edges of C to form a triangle, we have a
clique on the vertices of S as desired. To prove the claim, we augment the graph
G by adding a vertex vl connected to every vertex in S, and by adding a vertex v2
connected to every vertex in C. Because SI = 3 and ICI > 3, the augmented graph
is still 3-connected. Therefore there exist at least three vertex-disjoint paths from vl
to v2. Each internal vertex on each of these paths must be in G2, and each path must
contain an edge from v to a vertex of S and an edge from a vertex of C to v2. We
obtain the desired paths by removing v and v 2 from each path.
D
Theorem 2.8. For any graph G excluding a single-crossing graph H as a minor, we
can construct in O(n4 ) time a series of clique-sumoperationsG = G1&G2E... E Gm
where each Gi, 1 < i < m, is a minor of G and is either a planar graph or a graph
of treewidth at most CH. Here each 0 is a 0-, 1-, 2- or 3-sum.
43
Proof. The algorithm proceeds by recursively determining connectivity of subgraphs
of the original graph, where different decompositions are used depending on the type
of cut. When considering graph G, we first determine whether it is 1-, 2-, or 3connected.
If G is disconnected, each of its connected components is considered
separately, and all are joined by O-sums to form G. If G has a 1-cut, 2-cut, or strong
3-cut S, we recursively apply the algorithm on the augmented components induced
by S, applying, respectively, 1-, 2-, or 3-sums. Finally, if G is 3-connected but has no
strong 3-cut, then we claim that it is either planar or has treewidth at most
CH.
We first prove the correctness of the outline above, and later fill in the algorithmic
details and analyze the running time. To show that the recursive application of the
algorithm will yield a correct solution, we need to show that each subgraph created is
a minor of G (and hence is H-minor-free by Lemma 2.1). A connected component of a
disconnected graph or an augmented component resulting from a 1-cut is a subgraph
of G, and hence a minor. For an augmented component formed by a 2-cut, the 2connectivity of G guarantees that any component must connect to both vertices in the
cut, and hence edges can be contracted to yield the edge added in the augmentation.
Lemma 2.7 handles the case in which there is a strong 3-cut.
Finally, we need to prove that if the graph G is 3-connected but has no strong
3-cut, then either the treewidth of G is at most
CH
or G is planar. Suppose instead
that neither of these properties hold. Since G is H-minor-free and 3-connected, by
Theorem 2.6, G can be obtained by 3-sums of a sequence of graphs C, where each
graph in C is either planar or of treewidth at most
CH
> 4. As G has no strong
3-cut, for any join set S in the clique-sum decomposition G - S may contain at most
one component with more than one vertex, and hence at most one graph in C can
have more than four vertices. If every graph in C has at most four vertices and hence
treewidth at most 3 (and hence less than
treewidth less than
CH,
CH),
then by Lemma 2.4, G would have
contradicting our assumption.
We can thus conclude that
C contains subgraphs of K4 and one planar graph of at least five vertices and of
treewidth greater than
CH.
To complete the proof of correctness, we will show that our assumption about C
44
yields a contradiction, that is, that G has a strong 3-cut. Since G is not planar but
every graph in C is planar, during the clique-sum operations forming G, there exists a
graph J E C and a 3-sum G" = G' ® J with join set S such that G' is planar but G" is
not planar. It is not possible to form planar embeddings of both J and G' such that S
forms the outer face, since if this were possible, the two embeddings could be joined
to form a planar embedding of G" (e.g. J would be embedded inside the triangle
and G' outside). We can thus conclude that there are at least three components in
G" - S, which implies that S is a strong 3-cut in G" and hence in G, a contradiction.
To analyze the running time of the algorithm, we first recall that any graph G
excluding an r-clique as a minor cannot have more than (0.319+o(1))(r logr) IV(G)I
edges [161]. This implies that for any single-crossing-minor-free graph G, E(G)I =
o(IV(G)I).
To run the algorithm, we apply algorithms to obtain all connected components
and 1-cuts in linear time [158], all 2-cuts [114, 134], and all O(n 2 ) 3-cuts in O(n 2 )
time [115]. Checking whether a particular 3-cut is strong can be accomplished in O(n)
time using depth-first search. All other operations, including checking if a graph is
planar or has treewidth at most
CH,
can be performed in linear time [166, 32].
To set up recurrence relations, we make the assumption that for each cut we split
a graph into two 0-, 1-, or 2-connected components or into at most three 3-connected
components at a particular iteration. The running time of one iteration, excluding
recursive calls, is O(n). For T(n) the running time on an input of size n, for a O-sum
involving a component of size nl we obtain the equation:
T(n) = T(nl) + T(n - nl) + O(n), n
2.
The recursive calls for a 1-cut yield the following equation
T(n) = T(nl) + T(n - n + 1)+ O(n), n l
2
where nl and n - nl + I are the sizes of the two augmented components. Similarly,
45
for recursive calls for a 2-cut, we have
T(n) = T(nl) + T(n - nL
1 + 2) + O(n),
3.
n
For recursive calls for a strong 3-cut with exactly two components, we have
T(n) = T(nl) + T(n - n + 3) + O(n3), nl > 4.
Finally, if we have recursive calls for a strong 3-cut with at least three components,
we have
T(n) = T(nl)+T(n 2)+T(n-nl-n
3
2 +6)+O(n ),
4 <nl,n 2,n-nl-n
2 +6 < n-2
where n 1 , n 2 , and n - nl - n 2 + 6 are the sizes of the augmented components (the
last being the third and all subsequent components taken together). The additive
terms (+1, +2, +3, +6) are due to the duplication of the vertices of the cut in the
augmented components. Solving this recurrence gives a worst-case running time of
O(n4 ).
0
Even for excluded graphs H where
CH
is huge, the value of cH does not contribute
to the asymptotic complexity of the algorithm presented above. However, it does
contribute heavily to the constant hidden in the order notation. In some contexts,
it may make sense to replace Bodlaender's linear-time algorithm for determining
treewidth exactly with an approximation. Amir [11] gives an algorithm running in
time 0(24.38cHn2cH) which either returns a tree-decomposition of width at most 4 cH
or answers that the treewidth is more than
Theorem 2 with
CH
CH.
This can be used to prove a version of
replaced by 4 cH but whose dependence on
CH
is more reasonable.
Similar substitutions will be possible in our approximation algorithms, discussed in
Section 2.5.
46
2.3
Locally Bounded Treewidth of Single-Crossing-
Minor-Free Graphs
In this section we establish the locally bounded treewidth of single-crossing-minorfree graphs, which provides the structure on which the approximation schemes of
Section 2.5 are built. First we demonstrate how clique sums and local treewidth are
correlated. Next, we discuss layer decompositions and present a tree decomposition
algorithm for a subgraph induced by a sequence of consecutive layers.
2.3.1
Bounded Local Treewidth
Lemma 2.9. If G1 and G2 aregraphssuch that ltwGl (r) < f(r) and ltwG2(r) < f(r)
for a function f (r) > 0 for all r E N, and G = G1
k G2 ,
then ltwG(r) < f (r).
Proof. To show ltwG(r) < f(r), we prove that for any v E V(G) and for all r > 0,
tw(G[NE(v)]) < f(r).
We use W to denote the join set of G1
Ok
G2
and without
loss of generality, we assume v is from G1. As the claim holds trivially for r = 0, in
the remainder of the proof we assume r > 0. Moreover, since if NG(v) contains only
vertices originally from G1, the result follows from the fact that twGl'(r) < f(r), we
assume that NG(v) contains vertices from G2 that are not in W. We consider two
cases, depending on whether v E W.
If v E W, then N (v) C NG1(v) U N 2 (). In addition, since r > 1 and vertices of
W form a clique in Gi for i = 1,2, W C N ~ (v). Using these two facts, we conclude
that G[N (v)] is a subgraph of Gi[N
(v)] ®DG2 [N 2(v)] over the join set W. Thus,
by Lemmas 2.3 and 2.4, we know
tw(G[N (v)]) < max{tw(G [Nl (v)]), tw(G2[N2 (V)])}< f(r)
We now consider the case in which v
W. Each vertex in NG(v) is either in G 1,
and hence is in NG1 (v), or it is in G2, and hence is in N(v)
- W. To further describe
the latter set, we consider the distance between v and any vertex u in the set. Since
W is the set of vertices shared by G1 and G2 in G, at least one vertex of W is on
47
the shortest path from v to u in G and hence is at distance of at most r - 1 from
v, hence W n NN 1 (v) Z 0. We let p be the minimum distance between v and any
vertex in the set W n N~G (v) and observe that since v in not in W, we can conclude
that 1 < p < r - 1. We further observe that since each vertex u in NE(V) - W is at
distance at most r from v, it must be within distance r - p of some vertex of W, or
u E N2-P(W). Thus N (v) C NG1 (v) U NG;P(W).
To complete the proof, we use the characterization of N (v) as a subset of NG1 (v)U
NGP(W) to obtain an upper bound on its treewidth. First we show that G1 [N 1 (v)] E
G2[N~P(W)] can be formed using the join set W, as W is a subset of each of the
constituent graphs (each vertex of W is at distance at most r from v in G1 since at
least one vertex of W is at distance p < r -1 from v and vertices of W form a clique in
G1 ). Since N~(v) C NG1 (v)UN~P(W), as shown in the previous paragraph, G[NE(v)]
is a subgraph of G1[N 1l(v)] D G2 [NZP(W)], and hence we can apply Lemma 2.4 to
obtain the following result:
tw(G[N(v)]) < max{tw(Gl[N 1 (v)]),tw(G2 [N2P(W)])}.
(2.1)
By Lemma 2.3, since p > 1 clearly G2 [N~2P(W)] is a subgraph of G2 [N2 (W)], and
hence
tw(G 2 [N2P(W)]) < tw(G 2[N2 1 (W)])
Combining (2.1), (2.2), and the fact that tw(G1[N~(v)])
(2.2)
< f(r) (our assumption
about G 1), we obtain
tw(G[Nb(v)]) < max{f(r), tw(G2[N21(W))}-
(2.3)
Since W is a clique in G2, by Lemma 2.5,
tw(G2[N21 (W)]) < tw(G2[N
(v)]) < f(r).
(2.4)
Finally, as a consequence of (2.3) and (2.4), we conclude that tw(G[NE(v)]) < f(r),
48
Figure 2-2: The graph Vs.
as needed to complete the proof of the lemma.
o
Theorem 2.10 demonstrates our main result on the local treewidth of singlecrossing-minor-free graphs.
Theorem 2.10. For any single-crossing-minor-free
graphG excludinga single-crossing
graphH as a minor and for all r > 0, ltwG(r) < 3r + cH.
Proof. By Theorem 2.6, we can assume G = G1 ( G2 & ... DGm where each Gi,
1 < i < m, is either a planar graph or a graph of treewidth at most cH; we prove
the theorem by induction on m.
In the base case, if G1 is a planar graph then
ltwG(r) = twG' (r) = 3r - 1 < 3r + CH, CH > 0, as the treewidth of a r-outerplanar
graph is at most 3r - 1 [2]. If instead G1 has treewidth at most CH, then ltw G (r) =
1tw G '
(r)
= CH
< 3r +
CH,
r > 0. To prove the general case, we assume the induction
hypothesis is true for m = h, and we prove the hypothesis for m = h + 1 by setting
G' = G1
G2
e ...
D Gh and G" = Gh+1. By the induction hypothesis, ltwG'(r) <
3r + CH and ltwG"(r) < 3r + cH; by applying Lemma 2.9, we conclude that
ltwG'G" (r) < 3r + CH, as needed.
twG =
I
Using the fact that K5 and K3,3 are single-crossing graphs (Figure 2-1), we observe
that K5 -minor-free graphs and K3,3-minor-free graphs are single-crossing-minor-free
graphs. Although generalized by Theorem 2.8 for single-crossing-minor-free graphs,
for more precise results we rely on Wagner's characterizations [164]. He proved that
a graph is K3,3 -minor-free if and only if it can be obtained from planar graphs and
K5 by 0-, 1-, and 2-sums and that a graph is K 5 -minor-free if and only if it can be
obtained from planar graphs and V8 (the graph obtained from a cycle of length 8
by joining each pair of diagonally opposite vertices by an edge, shown in Figure 2-2)
49
by 0-, 1-, 2-, and 3-sums. Since both K5 and V8 have treewidth four, the value of
constant CH in the proof of Theorem 2.10 is four, and we have:
Corollary 2.11. If G is a K 5 -minor-free or K 3 ,3 -minor-free graph then ltwG(r)
<
3r + 4.
2.3.2
Local Treewidth and Layer Decompositions
To take advantage of the bound on local treewidth, we define a layer decomposition,
prove a bound on the treewidth of a subgraph induced on consecutive layers, and
then provide an algorithm that forms a tree decomposition of such a subgraph. The
concept of the kth outer face in planar graphs can be replaced by the concept of the
kth layer (or level) in graphs of locally bounded treewidth. The kth layer (Lk) of
a graph G consists of all vertices at distance k from an arbitrary fixed vertex
of
V(G). We denote consecutive layers from i to j by L[i,j] = Ui<k<jLk, and call such
a representation a layer decomposition.
Theorem 2.12. For any graph G excluding a single-crossing graph H as a minor,
the treewidth of G[L[i, j]] is bounded above by 3(j - i + 1) + cH.
Proof. By contracting the connected subgraph G[L[O,i-1]] to a vertex v' and applying
Lemma 2.1, we obtain another H-minor-free graph G'. As all vertices at distance d,
i < d < j, from v in G are at distance d', 1 < d' < j-i+1,
from v' in G' and all vertices
at distance more than j from v in G are at distance more than j - i + 1 from v' in G',
we have G[L[i,j]] = G'[L[1,j - i + 1]]. Thus tw(G[L[i, j]]) = tw(G'[L[1,j - i + 1]).
Since all vertices of L[1,j - i + 1] in G' are in the j - i + 1-neighborhood of v',
tw(G'[L[1,j - i + 1]]) < tw(G'[NG,i+1 (v')]). By the definition of local treewidth,
tw(G'[NGTi+l(v')]) < ltw G (j- i + 1). Finally by Theorem 2.10, we have ltwG'(j -
i + 1) < 3(j
-
i + 1) +
CH.
Using these facts, tw(G[L[i,j]]) < 3(j - i + 1) +
desired.
cH, as
o
Theorem 2.12 gives an upper bound on the treewidth of consecutive layers from
i to j, but does not provide a constructive algorithm to obtain a tree decomposition
of this width.
50
I P'.-r
-1
U,~
C'
G
Figure 2-3: The replacement of the part of path P between a and b by edge {a, b}.
Although we can construct a tree decomposition of width 3(j - i) +
CH
in linear
time using Bodlaender's algorithm [32], again the hidden constant factor will depend
on this entire width. Below we show how to reduce the constant to depend only on
CH,
permitting substitution of approximation algorithms as was done at the end of
Section 2.2.
Before stating the main theorem on construction of a tree decomposition of consecutive layers, we present a simple lemma.
Lemma 2.13. For G = G1 G2
.'
G,, if there exists a vertex v E V(G) such
that each vertex of G is at distance at most r from v, then in each Gi, 1 < i < m,
there exists a vertex vi such that each vertex of Gi is at distance at most r from vi.
Proof. We use induction on m, the number of Gi's. If m = 1, the basis of induction
is clearly true. We assume the induction hypothesis is true for m < h, and we prove
the hypothesis for m = h + 1. We suppose G = G' · G", with join set W, where
G' = G1
G ED... E
.2
Gh
and G" =
Gh+1.
In order to apply the induction hypothesis
to G' and G", we will need to find vertices v' and v" in G' and G" such that each
vertex of G' is at distance at most r from v' and each vertex of G" is at distance at
most r from v".
In order to apply the induction hypothesis to G' and G", it will suffice to show that
there is a path of length at most r from v to each vertex u in G' such that each vertex
of the path is in G', that is v = v' as defined in the previous paragraph (an analogous
argument can be used to show the existence of a path in G" to any vertex in G").
51
Suppose instead that every path of length at most r from v to u passed through at
least one vertex w in V(G") - V(G'), and consider one such path P. Clearly v and u
are not both in W, since otherwise there would exist an edge {v, u} in the clique with
vertex set W. Without loss of generality we assume u E V(G') - V(G"), and observe
that since the path from v to w and the path from w to u must pass through W, we
can define a and b to be the first and last vertices in W on the path in order from v
to u (see Figure 2-3), where possibly v = a or a = b (or both). We can then form a
new path P' consisting of the subpath from v to a, the edge {a, b} if a
$b
(which
must exist since the vertices of W form a clique in G'), and the subpath from b to u.
The path P' has length at most the length of P and is entirely in G', contradicting
]
our assumption and completing the proof.
We are ready to present our algorithm for construction of a tree decomposition
for a constant number of consecutive layers.
Theorem 2.14. For any single-crossing-minor-freegraph G, we construct a tree
decompositionfor G[L[i,j]] of treewidth3(j - i + 1) + CH in O((j - i + 1)3 n + n4 )
time; for a K 3 ,3-minor-free or Ks5 -minor-free graph G, the running time can be reduced
to O((j - i + 1)3 n) or O((j - i + 1)3 n + n2 ), respectively.
Proof. As in the proof of Theorem 2.12, we contract the connected subgraph G[L[O,i1]] to a vertex v' and obtain another single-crossing-minor-free graph G' such that
G[L[i, j]] = G'[L[1,j - i + 1]]. By Lemma 2.1, the graph G" = G'[L[O,j - i + 1]] is a
single-crossing-minor-free graph excluding the same H and by the definition of layers
each vertex in G" is at distance at most j - i + 1 from v'. By Theorem 2.8, we can
determine a set of clique-sum operations of graph G" in O(n4 ) time (improved to O(n)
for G K 3,3-minor-free using the result of Asano [20] and O(n
2
)
for G K 5-minor-free
using the result of K6zdy and McGuinness [118]).
After determining a set of clique-sum operations of G" = G1
G2
... * EDG,,
we
construct a tree decomposition for each Gi, 1 < i < m. If Gi is a graph of treewidth
at most
CH,
we can easily construct a tree decomposition of constant width in linear
time [32] (for the special cases, K 5 or V8, a constant time construction is possible).
52
We now consider the case in which Gi is a planar graph. By Lemma 2.13, in each Gi,
there exists a vertex vi such that each vertex in Gi is at distance at most j - i + 1
from vi. It is known that if a planar graph has a rooted spanning tree T in which the
longest path has length d, then a tree decompositionof the graph with width at most
3d can be found in time O(dn) [23, 86]. Since each vertex in Gi is at distance at most
j - i + 1 from vi, by breadth-first search we can construct a spanning tree rooted at
vi with the longest path of length at most j - i + 1. Hence we can construct a tree
decompositionfor Gi of treewidth 3(j - i + 1) in time O((j - i + 1) IV(Gi)l).
Having tree decompositions of Gi's, 1 < i < m, in the rest of the algorithm, we
glue together the tree decompositions of Gi's using the construction given in the proof
of Lemma 2.4. To this end, we introduce an array Nodes indexed by all subsets of
V(G") of size at most three. In this array, for each subset whose elements form a
clique, we specify a node of the tree decomposition which contains this subset. We
note that for each clique C in Gi, there exists a node z of TD(G") such that all vertices
of C appear in the bag of z [37]. This array is initialized as part of a preprocessing
stage of the algorithm. Now, for the D operation between G1 D ... () Gh and Gh+l
over the join set W, using array Nodes, we find a node a in the tree decomposition of
G1 O... ·
Gh whose bag contains W. Since we have the tree decomposition of Gh+l,
and W is a clique of size at most three that must appear in some bag in any tree
decomposition, we can find the node a' of the tree decomposition whose bag contains
W by brute force search over all subsets of bags of size at most three. Simultaneously,
we update array Nodes by subsets of V(G") which form a clique and appear in bags
of the tree decomposition of Gh+l. Then we add an edge between a and a'. As the
number of nodes in a tree decomposition of Gh+1 is O(IV(Gh+l)l) and each bag has
size at most 3(j - i + 1) (and thus there are at most O((j - i + 1)3 ) choices for a
subset of size at most three), this operation takes O((j - i + 1)3. IV(Gh+l)l) time for
Gh+l The claimed running time follows from the time required to determine a set
of clique-sum operations, the time required to construct tree decompositions, the
time needed to glue tree decompositions together and the fact that Eim1 IV(Gi)l =
53
O(IV(G")I). Here we note that the only difference between the running times for the
general algorithm and those for K 3,3 -minor-free or K 5 -minor-free graphs is the time
required to determine a set of clique-sum operations (O(n) time for the former graphs
and O(n 2 ) time for the latter graphs). The rest of the algorithm requires linear time
for all single-crossing-minor-free graphs.
Finally, we prove that the width of the constructed tree decomposition of G"
is 3(j
G1
-
G2
i + 1) +
e ... ·
CH.
We use induction on m, the number of Gi's, where G" =
Gm. For m = 1, G1 is either a planar graph of treewidth at most
3(j- i+ 1) or a graph of treewidth at most
CH.
In both cases the basis of the induction
is true. We assume the induction hypothesis is true for m = h, and we prove the
hypothesis for m = h + 1. For G = G1e G2 (D ..ED Gh, G" = G
Gh+l. By the
induction hypothesis, G and Gh+l each have treewidth at most 3(j - i + 1) +
CH.
We can then apply Lemma 2.4 to conclude that the treewidth of G" is also at most
3(j - i + 1) +
CH,
as needed to complete the proof.
E[
2.4 Approximating Treewidth
A large amount of effort has been put into determining treewidth, which is NPcomplete even if we restrict the input graph to graphs of bounded degree [38], cocomparability graphs [14, 108], bipartite graphs [121], or the complements of bipartite
graphs [14]. However, treewidth can be computed exactly in polynomial time for
chordal graphs, permutation graphs [36], circular-arc graphs [156], circle graphs [121],
distance-hereditary graphs [44], and for graphs of a fixed treewidth [32].
A constant factor approximation algorithm for treewidth of planar graphs is
known. This approximation algorithm is a consequence of the polynomial-time algorithm given by Seymour and Thomas [155] for computing the parameter branchwidth, whose value approximates treewidth within a factor of 1.5. Using the notions
of branchwidth and clique-sum decomposition, we demonstrate a polynomial-time
algorithm that approximates within a constant factor the treewidth of any singlecrossing-minor-free graph. Using a different approach, this result will be generalized
54
{b, f}
{f, d}
L
n
a
]~~~~~~~~~~~~~~~~~~~~~~
U,(:rK
{b,c}
b
{a, b}
Jf,d)
{a,
IJ , --I
f
e
1, -1
e
t,
{d,
e}
I I I
le. fl
I I- I
Figure 2-4: A graph and two branch decompositions of it. The first has width 4
and the second has width 3.
for all H-minor-free graphs, for a fixed H, in Chapter 9.
Analogous to the relationship between treewidth and tree decompositions, the
notion of branchwidth is related to a decomposition based on the edges. A branch
decomposition of a graph G is a pair (T, r), where T is a tree with vertices of degree
1 or 3 and r is a bijection from the set of leaves of T to E(G). The order of an edge
e in T is the number of vertices v E V(G) such that there are leaves t 1, t 2 in T in
different components of T(V(T), E(T) - e) with T(tl) and r(t2) both containing v as
an endpoint. The width of (T, r) is the maximum order over all edges of T, and the
branchwidth of G, bw(G), is the minimum width over all branch decompositions of
G (if IE(G)I < 1, we define the branchwidth to be 0; if
IE(G)I=
0, then G has no
branch decomposition; if IE(G)I = 1, then G has a branch decomposition consisting
of a tree with one vertex and the width of this branch decompositionis consideredto
be 0). It is well-known that, if H -< G or H -<c G, then bw(H) < bw(G). Figure 2-4
provides examples of branch decompositions.
The following result of Robertson and Seymour [145] relates branchwidth to
treewidth.
Theorem 2.15 ([145], Section 5). For any connected graph G where E(G)I > 3,
bw(G) < tw(G) + 1 <
bw(G).
We make use of an approximation algorithm for computing treewidth of planar graphs as one of two "base cases" in our algorithm for single-crossing-minorfree graphs. While it remains an open question whether there exists a polynomialtime constant-factor approximation algorithm for computing the treewidth of general
55
graphs, the branchwidth of a planar graph can be computed in polynomial time.
Theorem 2.16. ([155], Sections 7 and 9) One can construct an algorithm that, given
a planar graphG,
1. computes in O(n2 logn) time the branchwidth of G; and
2. computes in O(n4 ) time a branch decomposition of G with optimal width.
Theorem 2.17. One can construct an algorithm that, given a planar graph G,
1. computes in O(n2 log n) time a value k with tw(G) < k < tw(G); and
2. computes in O(n4 ) time a tree decomposition D of width k of G, where tw(G) <
k< tw(G) + 1.
Proof. The proof is straightforward using O(IE(G)12 ) algorithms of Robertson and
Seymour [145] which convert a branch decomposition of width at most b to a tree
decomposition of width at most b- 1, and convert a tree decomposition of width at
most k to a branch decomposition of width at most k + 1.
O
The main theorem of this section relies on Theorem 2.17 as well as clique-sum
decompositions.
Theorem 2.18. For any single-crossinggraphH, we can constructan algorithmthat,
given an H-minor-free graph as input, outputs in O(n 4 ) time a tree decomposition of
G of width k where tw(G) < k <
tw(G) + 1.
Proof. The algorithm consists of the following four steps:
Step 1: Let G be a graph excluding a single-crossing graph H. By Theorem 2.8, we
can obtain a clique-sum decomposition G = G1 EG 2
'...DGm where each Gi,
1 < i < m, is a minor of G and is either a planar graph or a graph of treewidth
at most CH. According to the same theorem, this step can be executed in O(n4 )
time. Let B be the set of the indices of the bounded treewidth components
and P be the set of planar components: B = {i
P= {1,...,m}-B.
56
1 < i < m, tw(Gi) < cH},
Step 2: By Theorem 2.17, we can construct, for any i E P, a tree decomposition
TD(Gi) of Gi with width ki and such that
tw(Gi)
ki <
tw(Gi) + 1
for all i E P.
(2.5)
The construction of each of these tree decompositions takes O(IV(Gi)l 4 ) time.
Because m < n and El<i<m IV(Gi) = O(n), (it is simple to prove this by
induction looking at the sizes of the components in the recurrences used in the
proof of Theorem 2.8) the total time for this step is O(n4 ).
Step 3: Using Bodlaender's algorithm [32], for any i E B, we can obtain a tree
decomposition of Gi with minimum width ki in linear time, where the hidden
constant depends only on
CH.
Combining (2.5) with the fact that tw(Gi) = ki
for each i E B, we obtain
)<
k i<tw(Gi)
3
+1
for all i E 1,
m.
(2.6)
Step 4: Now that we have tree decompositions TD(Gi) of each Gi, we glue them
together using the construction given in the proof of Lemma 2.4, as detailed in
Theorem 2.14. In this way, we obtain a tree decomposition of G that has size
k = max({k I 1
max{tw(Gi)
i < m.
Combining this equality with (2.6), we have
i = ,..., m})
k < 3max{tw(Gi)
i = 1 ... , m}
1(2.7)
To see that this step can be executed in O(n 4 ) time, we observe that as described
in Theorem 2.14, for each of the O(IV(G)I) nodes in the tree decomposition,
we execute a brute force search in the array Nodes, indexed by all O(IV(G)I 3 )
subsets of V(G) of size at most three.
Finally, we prove that the algorithm is a 1.5-approximation. By Lemma 2.4, we
have tw(G) < max{tw(Gi)
Ii
= 1,..., m.
By Theorem 2.8, each Gi is a minor
of G and therefore tw(Gi) < tw(G) (since the class of graphs of treewidth at most
57
k is minor-closed). Thus, tw(G) = max{tw(Gi) I i = 1,.
conclude that tw(G)
k<
..
,m} and from (2.7) we
tw(G) + 1 and the theorem follows.
a
Using the same approach as Theorem 2.18, one can prove a potentially stronger
theorem:
Theorem 2.19. If we can compute the treewidth of any planar graph in polynomial
time, then we can compute the treewidth of any single-crossing-minor-free graph in
polynomial time.
Proof. We use the polynomial-time algorithm for computing treewidth of planar
graphs in Step 2 of the algorithm described in the proof of Theorem 2.18.
[1
2.5
Polynomial-time Approximation Schemes
2.5.1
General Schemes for Approximation on Special Classes
of Graphs
A polynomial-time approximation scheme for a problem is a family {Ae} of algorithms, where A, is a (1 + ) approximation for the problem that runs in time
polynomial in the length of its input (for fixed ). Inherent in the design of many
polynomial-time approximation schemes for NP-complete graph problems is the re-
striction of inputs to graph classes that guarantee additional structural properties.
Early work in the area demonstrated the possibility of using planarity to obtain approximation schemes [132], later generalized to graphs without a fixed minor [9].
These approaches are impractical; a performance ratio of two for the independent set
problem is achievable only for planar graphs of at least 22400vertices [59].
Practical approximation schemes for planar graphs were developed by Baker [23],
who formed a decomposition of G into overlapping k-outerplanar subgraphs. For any
planar embedding, vertices can be put into layers by iteratively removing vertices on
the outer face of the graph: vertices removed at the ith iteration are assigned to layer
i. Since a k-outerplanar graph is decomposed into at most k layers, a k-outerplanar
58
subgraph can be formed by removing vertices with layer number congruent to i mod
k. Baker's technique immediately implies (1 + 1/k)-factor approximation algorithms
for many problems that can be solved exactly on k-outerplanar graphs (e.g. maximum
independent set, minimum dominating set, and minimum vertex cover), as it suffices
to solve the problem exactly on each of the k subgraphs (one for each value of i)
and return the best of the k results. Consider an optimal answer in the full graph.
Since the sets of removed vertices partition the graph, one of these sets removes at
most 1/k of the vertices in the answer, and the exact solution on the corresponding
subgraph will be a (1 + 1/k)-approximation to the optimal answer for the full graph.
Chen [56] later generalized Baker's approach to form approximation algorithms of
ratio 1 + 1/log n for problems on K 3,3 -minor-free graphs and K5 -minor-free graphs;
due to the types of layers formed, these results were exclusively for maximization
problems. Eppstein [87] showed that Baker's technique can be extended by replacing
bounded outerplanarity with bounded local treewidth. As with k-outerplanar graphs,
a wide range of NP-complete problems can be solved in linear time on graphs of
bounded local treewidth. The decomposition by deleting every kth face is replaced
by deleting every kth level of a breadth-first tree of G, keeping in mind the fact that
that the treewidth of the resulting graphs is a function of k.
2.5.2
Approximation Schemes for Single-Crossing-Minor-Free
Graphs
In this section we use the bound on local treewidth established in Theorem 2.10 to
obtain polynomial-time approximation schemes. Among NP-optimization problems,
we mainly focus on those problems which are also hereditary, namely, problems which
determine a property that if valid for an input graph is also valid for any induced
subgraph of the input. For a property 7r, the maximum induced subgraph problem
MISP(7r) is finding a maximum induced subgraph with the property; in the weighted
version (WMISP(7r)), the input graph has weights on its vertices and the goal is to
find a maximum weight induced subgraph with the property. For example, we might
59
search for an induced subgraph of maximum size that is chordal, acyclic, without
cycles of a specified length, without edges, of maximum degree r > 1, bipartite or a
clique [167]. For exact definitions of various NP-hard problems in this chapter, the
reader is referred to Garey and Johnson's seminal book [99].
Yannakakis has shown that for many natural hereditary properties ir, MISP(7r)
is NP-complete even when restricted to planar graphs [167]. Using the results of
Section 2.3, we obtain approximation algorithms for both maximization and minimization problems such as the maximum independent set problem, the minimum
vertex cover problem and the minimum dominating set problem on single-crossingminor-free graphs.
graph
Theorem 2.20. ForG a non-negativevertex-weightedsingle-crossing-minor-free
excludingH, k > 1 an integer, and Time,r(w, n) the nondecreasingworst-caserunning time of WMISP(Tr) over an n-vertex partial w-tree whose tree decomposition
is given, the maximization problem WMISP(7r) for a hereditary property r over G
admits a polynomial-time approximation scheme of ratio 1 + 1/k with worst-case
running time in O(k
improves to O(k
4 + k Time,(3(k - 1) + cH, IVI)). The running time
V1
IVI + k Time,(3(k
- 1) + 4, IVI)) for G K 3, 3-minor-free and
O(k IV12 + k Time ,(3(k - 1) + 4, IVI)) for G K 5 -minor-free.
Proof. Our algorithm proceeds by creating k subgraphs of G, solving the problem on
each of the subgraphs, and returning the best solution for any of the subgraphs as the
solution for all of G. We make use of the locally bounded treewidth of G in order to
specify layers from which the subgraphs are derived and to prove that each subgraph
has bounded treewidth.
Given an assignment of vertices to layers numbered 1, 2,... created by breadthfirst search (layer i is all vertices at depth i), we use Li,j to denote the consecutive
layers numbered (j - 1)k + i through jk + i - 2 for 1 < i < k and j > 0 where for
convenience a layer is defined to be empty when its number is not between zero and
the total number of layers. Furthermore, we let Li = Uj>oLi,j and Gi = G[Li]. As
neither Li,j nor Li,j+1 contains the layer numbered jk + i - 1 and all edges appear
60
between consecutive layers, there are no edges between Li,j and Li,j+l. Moreover, as
no vertices in layer i - 1 appear in Lh for h = i mod k, each vertex appears in exactly
k - 1 of the Li's or Gi's, a fact we will use later in the proof.
We next use the bound on the treewidth of each Gi to obtain Opti, the maximum
weighted solution of WMISP(7r) on each Gi, 1 < i < k. In particular, we construct
a tree decomposition of width 3(k - 1) +
CH
for each Gi by adding edges between
tree decompositions for each G[Li,j], which in turn can be formed in
(n4 ) time
using Theorem 2.14 (O(n) for K3 ,3-minor-free graphs or O(n 2 ) time for K5 -minorfree graphs). The fact that the graphs G[Li,j] are disjoint means that the process
of adding edges to form a single tree is straight-forward. As in Lemma 2.4, by the
definition of Time, (w, n) as a nondecreasing function, since I[V(Gi) < IV(G)I, Opti
can be determined in Time,(3(k-
1) +CH, IV(G)I). The running time is Time,(3(k-
1) + 4, IV(G)I) for G K3 ,3 -minor-free or K5 -minor-free.
Finally, we take Opt,, the solution with maximum weight among the Opti's, as our
solution for graph G, and prove the ratio bound by showing that weight(Opt) < k
Weight(Optm
- k1
or k weight(Optm) > (k - 1) weight(Opt), where Opt is the maximum weighted
solution on graph G. By observing that Optm > Opti for each value of i, we show
that k weight(Optm) > Ei
weight(Opti). Since 7ris hereditary, weight(Opti) >
weight(Opt n Li), and hence Ei=l weight(Opti) > i=l weight(Opt nLi). Finally,
we recall that each vertex appears in exactly k - I of the
i's, from which we can
conclude YEk=lweight(Opt n Li) = (k - 1) weight(Opt), as needed to conclude the
proof that k weight(Optm) > (k- 1) weight(Opt).
The claimed running time follows immediately from the time to construct the tree
decomposition, the time to solve WMISP(r) for each Gi, and the number of Gi's.
Corollary 2.21. For G a non-negative vertex-weightedsingle-crossing-minor-free
graph excludingH, the maximum independent set problem admits a polynomial-time
approximationscheme of ratio 1 + 1/k with running time O(k
running time improvesto O(k
2 3k . n)
23k
n + k n4 ). The
for G K 3 ,3 -minor-free and O(k 23k n + k . n2 )
for G K 5 -minor-free.
61
Proof. Using dynamic programming on a tree decomposition, this problem can be
solved in 0(2 W · n) time, over each n-vertex partial w-tree whose tree decomposition
is given [8]. Thus the result follows from Theorem 2.20 for Timer(w, n)= 0(2w - n).
It is worth mentioning that our result can be applied to NP-minimization problems, e.g., the minimum vertex cover problem and the minimum dominating set
problem. The main difference between these two results and the previous one is that
the previous construction avoided overlap between the various sets Li,j of consecutive
layers, in the minimization setting we need to enforce overlap in order to achieve the
approximation guarantee. The ideas of the proofs of Theorems 2.22 and 2.23 follow
ideas of Grohe [103] for general graphs of locally bounded treewidth, which are in
fact Baker's ideas for planar graphs. Since there are not too many new ideas in the
proofs of Theorems 2.22 and 2.23 below with respect to the proof of Theorem 2.20,
we omit the proofs and refer the reader to [72] for their detailed proofs.
Theorem
2.22 ([72]). For G a single-crossing-minor-free graph and any integer
k > 1, the minimum weighted vertex cover problem admits a polynomial-time approx-
imation scheme of ratio 1+ 1/k with worst-caserunning time O(k.23 k.n+k.n 4 ). The
running time improves to O(k
23k . n)
for G K3, 3 -minor-free and O(k . 2 3k n + k. n 2)
for G K 5 -minor-free.
Theorem 2.23 ([72]). For G a single-crossing-minor-freegraph and any integer
k > 1, the minimum weighteddominating set problem admits a polynomial-timeapproximation scheme of ratio 1+2/k with worst-case running time O(k. 43k n+k .n4).
The running time improves to O(k.4 3 k.n) for G K 3,3 -minor-free and O(k.43 k. n+k.n 2 )
for G Ks5 -minor-free.
graphs, there arepolynomial-timeapTheorem 2.24. For single-crossing-minor-free
proximation algorithmswhose solutions convergetoward optimal as n increasesfor
maximum independentset, minimum vertex cover and minimum dominatingset.
Proof. The running time of algorithms introduced in Corollary 2.21 and Theorems 2.22
and 2.23 is O(ckn + n 4) where k is a parameter and c is a constant. Now, by taking
62
k = log nl, we obtain polynomial-time approximation algorithms of ratio + 1/(log n)
(or 1 + 2/(logn) for dominating set). As both 1/(logn) and 2/(logn) decrease as n
increases, the solutions converge toward optimal as n increases.
l
Finally, it is worth mentioning that our results for single-crossing-minor-free graphs
in this section have been generalized to all apex-minor-free graphs by a result of Demaine and Hajiaghayi [66] who showed that apex-minor-free graphs have linear local
treewidth, i.e., the concepts of having bounded local treewidth and having linear local
treewidth are equivalent in minor-closed graph families.
2.6
Concluding Remarks
In this chapter, we introduced the class of single-crossing-minor-free graphs, which
contains K3 ,3-minor-free graphs and K5-minor-free graphs, generalizations of planar
graphs, and demonstrated structural properties which gave rise to new algorithms.
We showed that single-crossing-minor-free graphs have linear local treewidth and
demonstrated how to obtain a tree decomposition of a fixed number of layers. Algorithms obtained using these properties include both approximation algorithms (e.g. a
1.5-approximation algorithm for treewidth and many polynomial-time approximation
schemes) and fixed-parameter algorithms (that we demonstrate in Chapter 3).
Extensions to the structural results could include finding clique-sum characterizations of graphs such as graphs excluding a double-crossing graph (or a graph with a
bounded number of crossings) as a minor. For each such class, polynomial-time constructions of the decompositions could be used for further algorithmic development.
Notice that the algorithm of Theorem 2.8 in Section 2.2.2 can serve as a general
heuristic for the computation or approximation of treewidth in a graph when the
resulting 3-connected components without strong 3-cuts are graphs whose treewidth
can be computed or approximated efficiently. Additional approximation algorithms
might include those which determine graph properties other than treewidth. As a consequence of Theorem 2.16, treewidth is a 1.5-approximation on branchwidth, which
immediately implies the existence of a 2.25-approximation for branchwidth of single63
crossing-minor-free graphs. It might be possible to use a clique-sum decomposition
to obtain a better approximation or an exact algorithm for branchwidth. We believe
that by using an approach similar to that described by Kezdy and McGuinness [118],
it is possible to obtain a polylogarithmic parallel algorithm that constructs a series
of clique-sum operations as in Theorem 2.8; details remain to be worked out.
We suspect that Baker's approach can be applied to obtain practical polynomialtime approximation schemes for other problems, such as variations on dominating sets [2] that have been solved on k-outerplanar graphs or graphs of bounded
treewidth [46, 76, 2]; several results in this direction have been found recently [70].
By considering other NP-complete problems that have good algorithms for planar
graphs and graphs of bounded treewidth, we may be able to extend the range of
problems to which using clique-sum decomposition techniques may be applied; some
results in this direction have already been found [74].
64
Chapter 3
Exponential Speedup of
Fixed-Parameter Algorithms for
Single-Crossing-Minor-FreeGraphs
According to a 1998 survey book [112], there are more than 200 published research
papers on solving domination-like problems on graphs. Because this problem is very
hard and NP-complete even for special kinds of graphs such as planar graphs, much
attention has focused on solving this problem on a more restricted class of graphs. It
is well-known that this problem can be solved on trees [60] or even the generalization
of trees, graphs of bounded treewidth [159]. The approximability of the dominating
set problem has received considerable attention, but it is not known and it is not
believed that this problem has constant-factor approximation algorithms on general
graphs [22].
Downey and Fellows [83]introduced the concept of fixed-parameter tractability to
handle NP-hardness. Unfortunately, according to this theory, it is very unlikely that
the k-dominating set problem has an efficient fixed-parameter algorithm for general
graphs. In contrast, this problem is fixed-parameter tractable on planar graphs. Alber
et al. [2] demonstrated a solution to the planar k-dominating set in time 0( 4 6Vi4n).
Indeed, this result was the first nontrivial result for the parameterized version of an
NP-hard problem where the exponent of the exponential term grows sublinearly in
65
the parameter. Recently, the running time of this algorithm was further improved to
13
0(227Vi'n) [116]and O(215
k + nn33
k4 ) [93]. One of the aims of this chapter is to
generalize this result to nonplanar classes of graphs.
In this Chapter, similar to the approach of Alber et al., we prove that for a singlecrossing graph H, the treewidth of any H-minor-free graph G having a k-dominating
set is bounded by O(vk).
As a result, we generalize current exponential speedup
in fixed-parameter algorithms on planar graphs to other kinds of graphs by showing
how we can solve the k-dominating set problem on any class of graphs excluding a
single-crossing graph as a minor in time 0(49.55vin°(1)). The genesis of our results
lies in a result of Chapter 2 on obtaining the local treewidth of the aforementioned
class of graphs.
Using the solution for the k-dominating set problem on planar graphs, Kloks et
al. [50, 123, 107] and Alber et al. [2, 6] obtained
exponential
speedup
in solving
other problems such as vertex cover, independent set, clique-transversal set, kernels
in digraph and feedback vertex set on planar graphs. In this chapter we also show
how our results can be extended to these problems and many other problems such as
variants of dominating set, edge dominating set, and a collection of vertex-removal
problems.
This chapter is organized as follows. First, we introduce the preliminary definitions
used throughout this chapter in Section 3.1. In Section 3.2, we prove two general
theorems concerning the construction of tree decompositions of width O(V/i) for these
graphs, and finally we consider the design of fast fixed-parameter algorithms for them.
In Section 3.2, we apply our general results to the k-dominating set problem, and in
Section 3.3, we describe how this result can be applied to derive fast fixed-parameter
algorithms for many different parameters.
In Section 3.4, we prove some graph-
theoretic results that provide a framework for designing fixed-parameter algorithms
for a collection of vertex-removal problems. In Section 3.5, we give some further
extensions of our results to graphs with linear local treewidth. We end with some
conclusions and open problems in Section 3.6.
66
3.1
Background
3.1.1
Preliminaries
For generalizations of algorithms on undirected graphs to directed graphs, we consider
underlying graphs of directed graphs. The underlying graph of a directed graph H
is the undirected graph G in which V(G) = V(H) and {u, v} E E(G) if and only if
(u, v) E E(H) or (v, u) E E(H).
Let s be an integer where 0 < s < 3 and C be a finite set of graphs. We say
that a graph class G is a clique-sum class if any of its graphs can be constructed by a
sequence of i-sums (i < s) applied to planar graphs and graphs in C. We call a graph
clique-sum if it is a member of a clique-sum class. We call the pair (C, s) the defining
pair of g and we call the maximum treewidth of graphs in C the base of 5 and the
base of graphs in S. A series of k-sums (not necessarily unique) which generate a
clique-sum graph G are called a decomposition of G into clique-sum operations.
According to the (nonalgorithmic) result of [142], if g is the class of graphs excluding a single crossing graph H then g is a clique-sum class with defining pair (C,s)
where the base of g is bounded by a constant CH depending only on H. In particular,
if H = K 3, 3 , the defining pair is ({K 5 }, 2) and CH = 4 [164] and if H = K 5 then the
defining pair is ({V 8 }, 3) and CH = 4 [164].
We call a clique-sum graph class Ga-recognizable if there exists an algorithm that
for any graph G E 5 outputs in O(na) time a sequence of clique-sums of graphs of
total size O(IV(G)I) that constructs G. We call a graph a-recognizable if it belongs
in some ao-recognizable clique-sum graph class.
One of the ingredients of our results is the following constructive version of the
result in [142], which has been proved in Chapter 2.
Theorem
3.1 ([72]). For any graph G excluding a single-crossing graph H as a
minor, we can construct in O(n4 ) time a series of clique-sumoperations G = G1
G2
...
® Gm where eachGi, 1 < i < m, is a minor of G and is either a planar
graphor a graph of treewidthat most CH. Here each · is a 0-, 1-, 2- or 3-sum.
67
In the remainder of the chapter we assume that
CH
is the smallest integer for which
Theorem 3.1 holds. Notice that, according to the terminology introduced before, any
graph class excluding a single-crossing graph as a minor is a 4-recognizable cliquesum graph class. As particular cases of Theorem 3.1 we mention that K 3 ,3 -minor-free
graphs are 1-recognizable [20] and K 5 -minor-free graphs are 2-recognizable [118]. For
more examples of graph classes that can be characterized by clique-sum decompositions, see the work of Diestel [77, 78].
A parameterized graph class (or just graph parameter) is a family F of classes
{(i, i > 0} where Ui>0 .Fi is the set of all graphs and for any i > 0 , i C Fi+. Given
two parameterized graph classes F1 and F2 and a natural number y > 1 we say that
F1
-,
F2 if for any i
0,1 C
2
i.
In the rest of this chapter, we identify a parameterized problem with the parameterized graph class corresponding to its "yes" instances.
Theorem 3.2. Let 5 be an al-recognizable clique-sum graph class with base c and
let F be a parameterizedgraph class. In addition, we assume that each graph in g
s < 3. Suppose also that there exist two
can be constructed using i-sums where i
positive real numbers p1, 32 such that:
(1) For any k > O, planar graphs in Fk have treewidth at most 1V + P2 and such
a tree decomposition can be found in O(na2) time.
(2) For any k > 0 and any i < s, if G1 EDiG2 E Fk then G 1,G2 E Fk
Then, for any k > 0, the graphs in g n ,Fkall have treewidth < max{(/31V + p2,c)}
and such a tree decomposition can be constructed in O(nm a x' a1,a2}+ ()
Proof. Let G E G n Fk and assume that G = G1 D)G2
1
'
(
s
n) time.
Gm where each Gi,
i < m, is either a planar graph or a graph of treewidth at most c. We use
induction on m, the number of Gi's. For m = 1, G = G1 is either a planar graph
V + 32 or a graph of treewidth at most c.
that from (1) has treewidth at most 31I
Thus the basis of the induction is true for both cases. We assume the induction
hypothesis is true for m = h, and we prove the hypothesis for m = h + 1. Let
68
G' = G1 G2 ED ( Gh and G" = Gh+l. Thus G = G'® G". By (2), both G' and G"
belong in
Fk.
By the induction hypothesis, tw(G') < max{fl3ik +
(1) tw(G") < max{iv
2,
c} and from
+ 32, c}. The proof, for m = h + 1, follows from this fact
and Lemma 2.4.
To construct a tree decomposition of the aforementioned width, first we construct
a tree decomposition of width at most PIVk + p2 for each planar graph in O(n02)
time. We also note that using Bodlaender's algorithm [32], we can obtain a tree
decomposition of width c for any graph of treewidth at most c in linear time (the
hidden constant only depends on c). Then having tree decompositions of Gi's, 1 <
i < m, in the rest of the algorithm, we glue together the tree decompositionsof Gi's
using the construction given in the proof of Lemma 2.4. To this end, we introduce an
array Nodes indexed by all subsets of V(G) of size at most s. In this array, for each
subset whose elements form a clique, we specify a node of the tree decomposition
which contains this subset. We note that for each clique C in Gi, there exists a node
z of TD(G) such that all vertices of C appear in the bag of z [37]. This array is
initialized as part of a preprocessing stage of the algorithm. Now, for the ® operation
between G1E...
Gh and Gh+1over the join set W, using array Nodes, we find a node
a in the tree decomposition of G1
...
E Gh whose bag contains
W. Because we have
the tree decomposition of Gh+l, we can find the node a' of the tree decomposition
whose bag contains W by brute force over all subsets of size at most s of bags.
Simultaneously, we update array Nodes by subsets of V(G) which form a clique and
appear in bags of the tree decomposition of Gh+l. Then we add an edge between a
and a'. As the number of nodes in a tree decomposition of Gh+1 is in O(IV(Gh+l)1)
and each bag has size at most O(Vk) (and thus there are at most 0((vrk)8) choices
for a subset of size at most s), this operation takes O((vk)
V(Gh+l)I) time for Gh+l.
The claimed running time follows from the time required to determine a set of
clique-sum operations, the time required to construct tree decompositions, the time
needed for gluing tree decompositions together and the fact that Eim IV(Gi)I =
o
o(IV(G)I).
Notice that condition (2) of Theorem 3.2 is not necessary when G excludes a single69
crossing graph and F is closed under taking of minors. Indeed, from Theorem 3.1,
we have that in the sequence of operations G = G1 EfG2 d ... d Gm, each Gi is a
minor of G and therefore, if G E Fk then each Gi is also a member of Fk. We resume
this observation to the following.
Theorem 3.3. Let g be the class of graphs excluding some single-crossing graph H
as a minor and let F be any minor-closed parameterized graph class. Suppose that
there exist real numbers o > 4, 1 such that any planar graph in Fk has treewidth
at most max{P3i1 + 30, CH} and such a tree decomposition can be found in O(na).
Then graphs in
gn
Fk all have treewidth < 1v/ + o0and such a tree decomposition
can be constructed in O(nm xa (
"4 )})
time.
Theorem 3.4. Let 0 be a graph class and let F be some parameterized graph class.
Suppose also for some positive real numbers c, ul, a2, /1, /2, 6 the following hold:
(1) For any k
O0,the graphs in
gn
Fk all have treewidth < max{c,/3lvk
+ 32})
and such a tree decomposition can be decided and constructed (if it exists) in
O(na2) time. We also assume testing membershipin 5 takes O(nal) time.
(2) Given a tree decomposition of width at most w of a graph, there exists an algorithm deciding whether the graph belongs in Fk in O(6wn) time.
a
Then there exists an algorithmdecidingin O(6max{c)lr+2-m+n-
an input graphG belongsin
gn
x {a
1 c 2})
time whether
k.-
Proof. First, we can test membership in g in O(nal) time. Then we can apply the
algorithm from (1) and (assuming success) supply the resulting tree decomposition
to the algorithm from (2).
3.2
o
Fixed-Parameter Algorithms for Dominating
Set
In this section, we describe some of the consequences of Theorems 3.2 and 3.4 on
the design of efficient fixed-parameter algorithms for a collection of parameterized
70
problems where their inputs are clique-sum graphs.
A dominating set of a graph G is a set of vertices of G such that each of the rest
of vertices has at least one neighbor in the set. We represent the k-dominating set
problem with the parameterized graph class DS where D)Sk contains graphs which
have a dominating set of size < k. Our target is to show how we can solve the kdominating set problem on clique-sum graphs, where H is a single-crossing graph,
in time O(cVn 0 (1)) instead of the current algorithms which run in time O(cknO(l))
for some constant c. By this result, we extend the current exponential speedup in
designing algorithms for planar graphs [6] to a more generalized class of graphs. In
fact, planar graphs are both K 3,3 -minor-free and K 5-minor-free graphs, where both
K 3,3 and K 5 are single-crossing graphs.
According to the result of [116] condition (1) of Theorem 3.2 is satisfied for 1 =
15.6,
32
= 50, and a 2
=
1. Moreover,
from [93], condition
(1) is also satisfied for
0l = 9.55, 32= 0 and a 2 = 4.
The next lemma shows that condition (2) of Theorem 3.2 also holds.
Lemma 3.5. If G = G1 em G2 has a k-dominating set, then both G1 and G2 have
dominating sets of size at most k.
Proof. Let the k-dominating set of G be S and W be the join set of G 1Gk G2. W.l.o.g.
we show that G1 has a dominating set of size k. If S1 = SnV(G1) is a dominating set
for G1 then the result immediately follows, otherwise there exists vertex w E V(G 1)
which is dominated by a vertex v E V(G 2 ) - V(G 1 ). One can observe that all such
vertices w are in W. Because v E S, but v
'
S1, set S = Sl + {w} has at most k
vertices and because W is a clique in G 1, S is a dominating set of size at most k in
G1.
o
Let g be any a-recognizable clique-sum class. Now by applying Theorem 3.2 for
/1
= 9.55,
2
= 0, a1 = a, and a 2 = 4 we have the following.
Theorem 3.6. If g is an a-recognizable clique-sum class of base c, then any member
G of g with a dominatingset of size at most < k has treewidthat most max{c,9.55v}
and the corresponding tree decomposition of G can be constructed in O(nm x{a4}) time.
71
From Theorem 3.6, we get that condition (1) of Theorem 3.4 is satisfied for /1 =
9.55, 2 = 0,
2 = maxa, 4}, and a 2 = 4. The main result in [8] shows that for
the graph parameter DS condition (2) of Theorem 3.4 is also satisfied for 6 = 4. We
conclude with the following.
Theorem 3.7. There is an algorithmthat in 0(4955Vn
+
) ) time solves the
nmax(a,4
k-dominating set problem for any a-recognizable clique-sum graph of base c. 1
Corollary 3.8. There is an algorithm that solves the k-dominating set problemfor
any graph class excluding some single crossing graph as a minor in 0(49554n + n4 )
time.
For the special cases of K5 -minor-free graphs and K3 ,3-minor-free graphs, we may
apply Theorem 3.2 for P1 = 15.6, /32 = 50, and ~a
2 = 1 and derive the following.
Corollary 3.9. There is an algorithm that solves the k-dominating set problemfor
any Ks5-minor-free graph in 0(4156v+5°n + n2 ) time and for any K3,3 -minor-free
graph in O(415.6vk+50n)time.
3.3 Algorithms for Parameters Bounded by the
Dominating-Set Number
We provide a general methodology for deriving fast fixed-parameter algorithms in this
section. First, we consider the following theorem which is an immediate consequence
of Theorem 3.4.
Theorem 3.10. Let 5 be a graph class and let F1, F2 be two parameterized graph
classes where F1 -< F2 for some natural number y > 1. Suppose also that there exist
positive real numbersal, a 2, pi, /32,6 such that:
(1) For any k > 0, the graphs in
g n.
k all have treewidth <
3P1
V + /32 and such a
tree decomposition can be decided and constructed (if it exists) in O(nQ2) time.
We also assume testing membership in 5 takes O(n"') time.
1In the rest of this chapter, we assume that constants, e.g. c, are small and they do not appear
in the powers, because they are absorbed into the O notation.
72
(2) There exists an algorithm deciding whether a graph of treewidth < w belongs in
fl in O(6Wn)time.
Then
(1) For any k > 0, the graphs in g n .Fk all have treewidth at most /3 VLy+32 and
such a tree decomposition can be constructed in O(n02) time.
(2) There exists an algorithm deciding in 0(561V5+02 + nmax{a1la2}) time whether
an input graphG belongsin g n Fkl.
Proof. Consequence (1) follows immediately from the definition of H. Consequence
Eo
(2) follows from Theorem 3.4.
The idea of our general technique is given by the following theorem that is a direct
consequence of Theorems 3.6 and 3.10.
Theorem 3.11. Let F be a parameterized graph class satisfying the following two
properties:
(1) It is possible to check membership in
Fk
of a graph G of treewidth at most w in
0(6wn) time for some positive real numberJ.
(2) F
DS.
Then
(1) Any clique-sum graph G of base c in jFk has treewidth at most max{9.55V/- +
8,c}.
(2) We can check whether an input graph G is in Fk in O(6955vn
+ nmax{a' 4})
on
an a-recognizable clique-sum graph of base c.
In what follows we explain how Theorem 3.11 applies for a series of graph parameters. In particular, we explain why Conditions (1) and (2) are satisfied for each
problem.
73
3.3.1
Variants of the Dominating Set Problem
A k-dominating set with property I on an undirected graph G is a k-dominating set
D of G which has the additional property II and the k-dominating set with property
II problem is the task to decide, given a graph G = (V, E), a property II, and a
positive integer k, whether or not there is a k-dominating set with property HI. Some
examples of this type of problems, which are mentioned in [2, 159, 160], are the
k-independent dominating set problem, the k-total dominating set problem, the kperfect dominating set problem, the k-perfect independent dominating set problem
also known as k-perfect code and the k-total perfect dominating set problem. For each
HI,we denote the corresponding dominating set problem by DS'.
Another variant is the weighted dominating set problem in which we have a graph
G = (V, E) together with an integer weight function w : V
N with w(v) > 0 for
all v E V. The weight of a vertex set D C V is defined as w(D) = EvED
w(v).
A k-weighted dominating set D of an undirected graph G is a dominating set D of
G with w(D) < k. The k-weighted dominating set problem is the task of deciding
whether or not there exits a k-weighted dominating set. We use the parameterized
class WDS to denote the k-weighted dominating set problem.
Condition (1) of Theorem 3.11 holds for 6 = 4 because of the following.
Theorem 3.12 ([2]). If a tree decomposition of width w of a graph is known, then a
solution to DSrn or to WDS can be determined in at most 0(4w · n) time.
Clearly, DSn
l DS
) and WVDS 1 DS and Condition (2) also holds. Therefore
Theorem 3.11 holds for y = 1 and 6 = 4 for DSnI and WDS.
Another related problem is the Y-domination problem (DSY) introduced in [25].
Definition 3.13. Let Y be a finite set of integers. A Y-domination is an assignment
f : V -- Y such that for each vertex x, f(N[x]) =
EveN[x] f(x)
1 where N[x]
stands for the neighborhood of x including x itself. An efficient Y-domination is an
assignment f with f(N[x]) = 1 for all vertices x E V. The value of a Y-domination f
is l{x f(x) > O)}. The weight of a Y-domination is CEv
f(x).
Two Y-dominations
are equivalent if they have the same closed neighborhood sum at every vertex. The
74
Y-domination problem asks whether the input graph G has an efficient Y-domination
of value at most k.
Using the generalized dynamic programming approach, Cai and Kloks [122]presents
an algorithm which runs in time O(IYlwn) to decide whether a graph G of treewidth
at most w has an efficient Y-domination of value at most k. It is worth mentioning
that, according to Bange et al. [25], a graph G has an efficient Y-domination if and
only if all equivalent Y-dominations have the same weight, and thus there is no need
to worry about the actual weight of an efficient Y-domination. Therefore, we have
that Condition
(1) of Theorem 3.11 holds for 6 =
IYI.
One can easily see that for Y-domination f of a graph G = (V, E), D = {xlf(x) >
O} is a dominating set, because each vertex x has at least one vertex with a positive
number assigned to it in N[x]. Thus if f is a Y-domination of G with value at most
k, then G also has a dominating set of size k. Therefore, DSY
~-1
DS and Condition
(2) holds as well. Theorem 3.11 applies for y = 1 and 6 = IYI.
3.3.2
Vertex Cover
The k-vertex cover problem (VC) asks whether there exists a subset C of at most k
vertices such that every edge of G has at least one endpoint in C. This problem is
one of the most popular problems in combinatorial optimization.
A great number of researchers believe that there is no polynomial time approximation algorithm achieving an approximation factor strictly smaller than 2 - e, for
a positive constant a, unless P = NP.
Currently, the best known lower bound for
this factor is 1.36 [82] and the best upper bound is 2 which can be obtained easily. The best current fixed-parameter tractable algorithm has time 0(1. 2 7 1k + kIVI)
[55]. In this section, we present an exponentially faster algorithm for this problem on
clique-sum graphs.
Without loss of generality, we can restrict our attention to graphs with no vertex
of degree zero. One can observe that if a graph G has a vertex cover of size k, then it
has also a k-dominating set. Therefore VC <1 DS and condition (1) of Theorem 3.11
75
holds. Moreover, Condition (2) holds because we can solve the vertex cover problem in
time 0(2W) if we know the tree decomposition of width w of a graph G [6]. Therefore,
Theorem 3.11 applies for y = 1 and 6 = 2 for the k-vertex problem.
A simple standard reduction to the problem kernel due to Buss and Goldsmith
[47] is as follows: Each vertex of degree greater than k must be in the vertex cover
of size k, because otherwise, not all edges can be covered. Thus we can obtain a
subgraph G' of G which has at most k 2 edges and at most k2 + k vertices and k' is
obtained from k reduced by the number of vertices of degree more than k. Chen et al.
[55] showed that in Buss and Goldsmith's approach one can even obtain a problem
kernel with at most 2k vertices in O(nk + k3 ) time. Thus, using this result with the
consequence (2) of Theorem 3.11 for VC, we obtain the following result.
Theorem 3.14. There exists an algorithm which decides the k-vertex cover problem
' ( a 4})
in 0(29'55Vk + kn + k3 + nm{
3.3.3
time on an a-recognizable clique-sum graph.
Edge Dominating Set
Another related problem is the edge dominating set problem £DS that given a graph
G asks for a set E' C E of k or fewer edges such that every edge in E shares at least
one endpoint with some edge in E'. Again without loss of generality we can assume
that graph G has no vertex of degree zero.
One can observe that if a graph G has a k-edge dominating set E', we can obtain a
vertex cover of size 2k by including both end-points of each edge e E E'. This means
that £DS 42 VC. In the previous section we showed that VC
1 DS therefore, the
Condition (2) of Theorem 3.11 holds for £DS when -y = 2. Condition (1) holds
because the edge dominating set problem can be solved in csn
[29, 23] (where ceds
is a small constant) on a tree decomposition of width w for a graph G. We conclude
that Theorem 3.11 applies for y = 2 and 6 = ceds.
Theorem 3.15. We can find a k-edge dominating set in O(ced,9'55n
time on an a-recognizable clique-sum graph.
76
+ nmax( a 4})
3.3.4
Clique-Transversal Set
A clique-transversal set of a connected graph G is a subset of vertices intersecting all
the maximal cliques of G [24, 49, 13, 106]. Because the vertex cover problem is NPcomplete even restricted to triangle-free planar graphs [50, 163], the clique-transversal
problem remains NP-complete on clique-sum graphs. The k-clique transversal problem CT asks whether the input graph has a clique-transversal set of size < k.
If a graph G has a k-clique-transversal, then it has a dominating set of size at
most k, because every vertex of G is contained in at least one maximal clique. This
implies that CT
j DS and Condition (2) of Theorem 3.11 holds for
= 1. Using
the general dynamic programming technique, we can solve the k-clique-transversal
problem on a graph G of treewidth at most w in O(c-n) for some constant ct. (The
approach is very similar to Chang et al. [50].) Therefore, Theorem 3.11 applies for
y = 1 and
= cct,.
Theorem 3.16. We can find a k-clique-transversalset in O(cd955n
+ nmax{a,4})
time on an a-recognizable clique-sum graph.
3.3.5
Maximal Matching
A matching in a graph G is a set E' of edges without common endpoints. A matching
in G is maximal if there is no other matching in G containing it. The k-maximal
matching problem MM
asks whether an input graph G has a maximal matching of
size < k.
Let E' be the edges of a maximal matching of G. Notice that the set of endpoints
of the edges in E' is a dominating set of G. Therefore MM
42
DS and the Condition
(2) of Theorem 3.11 holds. Condition (1) holds because the problem can be solved
in cmn [29] on a tree decomposition of width w for a graph G. Hence Theorem 3.11
gives the following result.
Theorem 3.17.
(1) Any clique-sum graph of base c with a minimum maximal marching of size k has
treewidth < 9.55V'
+ max{8, c}.
77
(2) One can decide whether an a-recognizable clique-sum graph G has a minimum
maximal matching of size at most k in O(c955vn + nmaa,4}) time.
3.3.6
Kernels in Digraphs
A set S of vertices in a digraph D = (V, A) is a kernel if S is independent and every
vertex in V - S has an out-neighbor in S. It has been shown that the problem of
deciding whether a digraph has a kernel is NP-complete [99]. Franenkel [95] showed
that the kernel problem remains NP-complete even for planar digraphs D with indegree and outdegree at most 2 and total degree at most 3. The k-kernel problem
1CER asks whether a graph has a kernel of size k. Moreover, we define the co-CER
problem as the one asking whether an n-vertex graph has a kernel of size n - k.
Here, we again observe that if a digraph D has a kernel of size at most k, then
its underlying graph G has a dominating set of cardinality at most k. Also for a connected digraph D = (V, A) and kernel K, V - K is a dominating set in the underlying
graph of D. Resuming these two facts we have CEZ <1 OS and co-K8£R <1 DS
and Condition (2) of Theorem 3.11 holds for both problems. We note that a slight
variation of Condition (1) also holds because Gutin et al. [107] gives an 0(3wkn)
time algorithm solving the k-kernel problem on graphs of treewidth at most w using the general dynamic programming approach. The straightforward adaptation of
Theorem 3.11 to this variation of Condition (1) gives the following.
Theorem 3.18.
(1) Any clique-sum graph of base c that has a kernel of size k or n - k has treewidth
< 9.55vk + max{8, c}.
(2) One can decide whether an a-recognizable clique-sum graph G of base c has a
kernel of size k in 0(3955Vnk + nmax{'
4)})
time.
(3) One can decide whether an a-recognizable clique-sum graph G of base c has a
kernel of size n - k in 0(39.554'n(n - k) + nma{a
78
4})
time.
3.4
Fixed-Parameter Algorithms for Vertex-Removal
Problems
In this section, we present general results allowing the construction of O(can)-time
algorithms for a collection of vertex-removal problems. To this end, we start with
some definitions. For any graph class 5 and any nonnegative integer k the graph class
k-almost(9) contains any graph G = (V, E) where there exists a subset S C V(G) of
size at most k such that G[V - S] E 5. We note that using this notation if 5 contains
all the edgeless graphs or forests then k-almost(g) is the class of graphs with vertex
cover < k or feedback vertex set < k.
We define Tr to be the class of graphs with treewidth < r. It is known that, for
1 < i < 2, T is exactly the class of Ki+2-minor-free
graphs (see e.g. [33]). We now
present a series of consequences of Theorem 3.3 for solving a collection of vertexremoval problems on classes of graphs excluding a single-crossing graph as a minor.
First, we need the following combinatorial lemma.
Lemma 3.19. Planar graphs in k-almost(T 2 ) have treewidth < 9.55xVk. Moreover,
such a tree decomposition can be found in 0(n 4 ) time.
Proof. Our target is to prove that planar graphs in k-almost(T2 ) are subgraphs of
planar graphs in
D Sk
and the result will follow from the fact that from [93], condition
(1) of Theorem 3.2 is also satisfied for
1 = 9.55, P2 = 0 and a2 = 4.
Let G be a planar graph and S be a set of < k vertices in G where G[V - S] is
K 4 -minor-free. Using Lemma 2.4, we can assume that G is a biconnected graph. In
addition, because k-almost(T 2) is a minor closed class, we can assume that G does not
have a 2-cut (a cut of size 2). In fact, if G has a 2-cut {u, v}, each of the connected
components of G - {u, v} plus edge {u, v} is a minor of G and thus by induction, we
can assume that they satisfy the conditions of the theorem. Then using Lemma 2.4,
we can glue the corresponding tree-decompositions together and obtain the desired
result for G. All these operations take at most 0(n 3 ) time.
Therefore, in the rest of the proof we assume that G does not have 1- or 279
cuts.
A consequence of this is that all the vertices of G have degree at least 3.
Another consequence is that two faces of G can have in common either a vertex or
an edge (otherwise, a 2-cut appears). Consider any planar embedding of G. We call
a face of this embedding exterior if it contains a vertex of S, otherwise we call it
interior. For each exterior face choose a vertex in S and connect it with the rest of its
vertices. We call the resulting graph H and we note that (a) G is a subgraph of H, (b)
H[V - S] = G[V - S], and (c) all the vertices of the exterior faces of H are dominated
by some vertex in S. We claim that S is a dominating set of H. Suppose, towards a
contradiction, that there is a vertex v that is not dominated by S. From (c) we can
assume that all of the faces containing v are interior. Let H' be the graph induced
by the vertices of these faces. As they are all interior, H' should be a subgraph of
H[V - S]. Let (x 1,...
,
,q,
x1 ) be a cyclic order of the neighbors of v and notice that
q > 3. Let also Fi be the face of H containing the vertices xi, v,
nt(i),
1 < i < q,
where next(i) = (i + 1) mod q + 1. We note that all these faces are pairwise distinct
otherwise v will be a 1-cut for H and G. Let Pi be the path connecting xi and
Xnt(i)
in H' avoiding v and containing only vertices of Fi. Recall now that two faces of H
have either v or an edge containing v in common. Therefore, it is impossible two
paths Pi, Pj,i
$
j, share an internal vertex. This implies that the contraction of
all the edges but one of each of these paths transforms H' to a wheel Wq that, as
q > 3, can be further contracted to a K4 (a weel Wq is the graph constructed taking
a cycle of length q and connecting all its vertices with a new vertex v). As H' is a
subgraph of the graph H[V - S] (b) implies that G[V - S] contains a K 4 , and this is
a contradiction. As now S is a dominating set for H the treewidth of H is at most
9.55v/.
From (a) we have that G is a subgraph of a planar graph in DSk and this
o
completes the proof of the theorem.
As mentioned before, k-almost(T 2) is a minor closed graph class. Moreover, if
O C T2, then for any k, k-almost(O) C k-almost(T2). Using now Theorem 3.3 we
conclude the following general result.
Theorem 3.20. Let 0 be any class of graphs with treewidth < 2 and let g the class of
80
graphs excluding some single-crossing graph H as a minor. Then the following hold.
(1) For any k > , all graphs in k-almost(CO) that also belong to g have treewidth
< max{9.55v,
cH}.
Moreover, the corresponding tree decomposition can be
found in O(n4 ) time.
(2) Suppose also that there exists an O(6Wn) algorithm that decides whether a given
graph belongs in k-almost (0) for graphs of treewidth at most w. Then, one can
decide whether a graph in 9 belongs in k-almost ()
in 0(6955kn + n4 ) time.
If {O1,...,0,Or} is a finite set of graphs, we denote by minor-excl(O 1 ,..., Or) the
class of graphs that are Oi-minor-free for all i = 1, . .., r.
As examples of problems for which Theorem 3.20 can be applied, we mention the
problems of checking whether a graph, after removing k vertices, is edgeless ( = To),
or has maximum degree < 2 (
= minor-excl(K, 3)), or becomes a a star forest
(9 = minor-excl(K3 , P3 )), or a caterpillar ( = minor-excl(K3 , subdivision of K1 ,3 )),
or a forest ( = T), or outerplanar ( = minor-excl(K4 , K2 ,3 )), or series-parallel,or
has treewidth < 2 ( =
2)-
We consider the cases where 9 = To and 9 = Tx in the next two subsections.
3.4.1
Feedback Vertex Set
A feedback vertex set (FVS) of a graph G is a set U of vertices such that every cycle
of G passes through at least one vertex of U. The previous known fixed-parameter
algorithms for solving the k-feedback vertex set problem had running time 0((2k +
1)kn 2 ) [83]and alternatively time 0((917k 4 )!(n+m)) [30] (m is the number of edges.)
Also there exists a randomized algorithm which needs O(c4kkn) time with probability
at least
-
(1 - )c4k [27]. The k-feedback vertex set problem (VS)
asks whether
an input graph has a feedback vertex set of size < k.
Kloks et al. [123] proved that the feedback vertex set problem on planar graphs
of treewidth at most w can be solved in O(cfsn) time for some constant Cbs. The
complexity of their algorithm is based on the fact that the number of edges of a
81
planar graph is bounded by a simple linear function of its vertices (i.e. 3n - 6). As
we have similar bound 3n- 5 for K 3 ,3 (K 5 )-minor-free graphs [20, 118], one can easily
observe that the algorithm of [123] works also for the more general case. Therefore,
Theorem 3.20 can be applied for g = T1 and
= cs and we have the following.
Theorem 3.21. If g is a graph class excluding some single-crossing graph H as a
minor then
(1) If G has a feedback vertex set of size at most k then G has treewidth at most
max{9.55x/k,cH}.
(2) We can check whether some n-vertex graph in 5 has a feedback vertex set of size
< k in O(cf55"'n+ n4 ) time.
Theorem 3.21 generalizes the results of [123]to any class of graphs excluding some
single-crossing graph H as a minor.
3.4.2
Improving Bounds for Vertex Cover
Alber et al. [6]proved that planar graphs in VCk have treewidth at most 4Fvvk-+5 <
6.93.x/ + 5. An easy improvement of this result is the folowing:
Lemma 3.22. If a planar graph has a vertex cover of size < k then its treewidthis
bounded by 5.52xv
Proof. Again using Lemma 2.4, we may assume that G is a biconnected graph. Let
S be a vertex cover in G where ISI < k. Consider a planar embedding of G.
Construct a triangulation H of G as follows: for any face F we add edges connect-
ing only vertices of F n S. This operation constructs a triangulation as there is no
pair of vertices in F - S that are consecutive in F. Moreover, as all the added edges
have endpoints in S, S is a vertex cover of H. We will prove that tw(H) < 5.52./'k
We may assume that H is a triangulation without double edges. To see this,
consider two edges e and e2 connecting vertices x and y and apply Lemma 2.4 on
the graphs Gin and Gex induced by the vertices included in each of the closed disks
bounded by the cycle where the two edges of this cycle are identified.
82
Notice now that for each vertex v E V(H) - S, all its neighbors are members of
S. This means that IV(H) - SI < r where r is the number of faces of J = H[S].
As H has not double edges, neither J has and therefore IE(J)I < 31V(J)I - 6. It is
knownthat r < E(J)I - IV(J)I + 2 and we get that r < 21V(J)I - 4. We conclude
that V(H)I < IV(J)I + 21V(J)I - 4 = 31SI-4
= 3k - 4. From [92], we know
that any n-vertex planar graph has treewidth at most
tw(H) < 9
3k- -4 < 2--vx
-2Vfi
f.
This means that
. As G is a subgraph of H and 2-v35 < 5.52, the
result follows.
°
Applying Theorem 3.20, we have that Condition (1) of Theorem 3.4 holds if F
is the class of graphs with vertex cover < k and g is any graph class excluding
some single-crossing graph H as a minor when c = CH, al = 4, a2
=
4, /1 = 5.52,
and P2 = 0. Also, as we mentioned in Subsection 3.3.2 it is possible to decide in
0(2wn) time if a graph has a vertex cover of size at most k. Therefore, Condition
(2) holds for
a
= 2. Concluding, we have the following improvement of the results
of Subsection 3.3.2 for any graph class excluding some single-crossing graph H as a
minor.
Theorem 3.23. If 9 is some graph class excluding some single-crossing graph H as
a minor then the following hold.
(1) If G E
has a vertex cover of size at most k then G has treewidth at most
max{5.52VT,cH}.
(2) There is an algorithm which checks whether some graph G E 9 has a vertex cover
of size < k in 0(2552V"k+ kn + k3 + n4 ) time.
Because E£ZS 2 VC, we can also obtain an O(ceds552Vn + n 4 )-time algorithm
for the edge dominating set problem on graphs excluding some single-crossing graph
as a minor.
83
3.5 Further Extensions
In this section, we obtain fixed-parameter algorithms with exponential speedup for
k-vertex cover and k-edge dominating set on classes of graphs that are not necessarily
classifiable as single-crossing minor-free graphs. Our approach, similar to the Alber
et al.'s approach [6], is a general one that can be applied to other problems.
Here we generalize the concept of layerwise separation, introduced by Alber's et
al. [6]for planar graphs, to general graphs.
Definition 3.24. Let G be a graphlayeredfrom a vertex v, and r be the numberof layers. A layerwise separation of width w and size s for G is a sequence (S1, S 2 ,.
of subsets of V, with property that Si C Uji-'i)
Lj; Si separates layers Lil
, Sr)
and
Li+; and r=,-1Sjl < s. Here we let Si = 0 for all i < 1 and i > r.
Now we relate the concept of layerwise separation to parameterized problems.
Definition 3.25. A parameterizedproblem P has LayerwiseSeparation Property
(LSP) of width w and size-factor d, if for each instance (G, k) of the problem P,
graph G admits a layerwise separation of width w and size dk.
For example, we can obtain constants w = 2 and d = 2 for the vertex cover
problem. In fact, consider a k-vertex cover C on a graph G and set Si = (LiULi+i)nC.
The Si's form a layerwise separation. Similarly, we can get constants w = 2 and d = 4
for the edge dominating set problem (see Alber et al. [6] for further examples).
Lemma 3.26. Let P be a parameterized problem on instance (G, k) that admits a
problem kernel of size dk. Then the parameterized problem P on the problem kernel
has LSP of width 1 and size-factor d.
Proof. Consider the problem kernel (G', k') for an instance (G, k) and obtain layering
L' for G' from arbitrary vertex v. Then clearly the sequence Si = Li for i = 1,.
,r
(r' is the number of layers), is a layerwise separation of width I and size k' < dk for
G'.
D
84
In fact, using Lemma 3.26 and the problem kernel of size 2k (see Subsection 3.3.2)
for the vertex cover problem, this problem has the LSP of width I and size-factor 2.
Now we are ready to present the main theorem of this section.
Theorem 3.27. Supposefor a graphG from a minor-closedclass of graphs,ltw(G) <
cr + c' and a tree decomposition of width ch + c' can be constructed in time f(n, h)
for any h consecutive layers. Also assume G admits a layerwise separation of width
w and size dk. Then we have tw(G) < 2v/-'d
+ cw + c'. Such a tree decomposition
can be computed in time O(kf(n, Vk-)).
Proof. The proof is very similar to the proof of Theorem 15 of Alber et al.'s work
[6] and for the sake of brevity we only mention the differences and omit the lengthy
details. In the proof, the concept of the kth outer face in planar graphs will be replaced
by the concept of the kth layer (or level) in graphs of locally bounded treewidth. More
precisely, Alber et al. [6]use the fact that treewidth of an h-outerplnar graph is 3h- 1,
but in our proof we use the fact that, for any graph G, treewidth of any h consecutive
layers is at most ch + c' [103, 72]. In addition, as mentioned before, Eppstein [87]
showed that a minor-closed graph class £ has bounded local treewidth if and only if
£ is H-minor-free for some apex graph H. (A simpler proof of this theorem can be
found in [65].) By Thomason [161], we know that any graph G excluding an r-clique
as a minor cannot have more than (0.319 + o(1))(rVlo-g) V(G)I edges. This implies
that for graph G mentioned in the statement of the theorem, E(G)I = O(IV(G)I),
similar to the corresponding relation for planar graphs. This fact is used for analyzing
El
the running time.
Corollary 3.28. For any H-minor-free graphG, whereH is a single-crossinggraph,
that admits a layerwiseseparationof widthw and size dk, we havetw(G) < 2v6
+
3w + cH. Such a tree decomposition can be computed in time O(k 5 / 2 n + kn 4 ). Furthermore, for any K3 ,3 (K 5 )-minor-free graph G, that admits a layerwise separation
of width w and size dk, we have tw(G) < 2v/dk+ 3w+4.
can be computed in time O(k 5 / 2n) (O(k 5 / 2n + kn 2 )).
85
Such a tree decomposition
Proof. The proof follows directly from Theorem 3.27 and the fact that for any singlecrossing-minor-free graph G, we can construct a tree decomposition of width 3h + ch
for any h consecutive layers in O(h3
n + n 4 ) time; for a K3 ,3-minor-free or K 5 -
minor-free graph G, the running time can be reduced to O(h 3 n) or O(h 3 n + n 2 ),
respectively [109, 72].
(CH =
4 for these graphs.)
0
Finally, we have this general theorem.
Theorem 3.29. Suppose for a graph G from a minor-closed class of graphs, ltw(G) <
cr+c'. Let P be a parameterized problem on G such that P has the LSP of width w and
size-factor d and there exists an 0(6wn)-time algorithm, given a tree decomposition
of width w for G, decides whether problem P has a solution of size k on G.
Then there exists an algorithm which decides whether P has a solution of size k
on G in time 0(2(11/3)(2V/2
0
-k+cw+c')n3.
1 + 63.698(2V/2k++cw')n).
Proof. The proof follows from Theorem 3.27, the fact that for graph G, treewidth of
any h consecutive layers is at most ch+ c' [103, 72], and finally the result of Amir [11],
which says for any graph G of treewidth w, we can construct a tree decomposition of
width at most (11/3)w in time 0(23.6 9 8 wn 3.01).
0
For example, Theorem 3.29 gives an exponential speed up, i.e., an algorithm with
running time 0(2°(NV)k301 + kn+ k3 + n4 ) (because c = 0(g) [87]), for solving vertex
cover on graphs of bounded genus.
Recently, it was established that all minor-closed classes of graphs with bounded
local treewidth, i.e., all minor-closed graph classes excluding an apex graph, in fact
have linear local treewidth [66]. Therefore Theorem 3.29 applies generally to any such
class of graphs.
3.6
Concluding Remarks
In this chapter, we considered H-minor-free graphs, where H is a single-crossing
graph, and proved that if these graphs have a k-dominating set then their treewidth
is at most c
for a small constant c. As a consequence, we obtained exponential
86
speedup in designing FPT algorithms for several NP-hard problems on these graphs,
especially K3 ,3-minor-free or K5 -minor-free graphs. In fact, our approach is a general
one that can be applied to several problems which can be reduced to the dominating
set problem as discussed in Section 3.3 or to problems that themselves can be solved
exponentially faster on planar graphs [6]. Here, we present several open problems
that are possible extensions of results of this chapter.
One topic of interest is finding other problems to which the technique of this
chapter can be applied. Moreover, it would be interesting to find other classes of
graphs than H-minor-free graphs, where H is a single-crossing graph, on which the
problems can be solved exponentially faster for parameter k. A partial answer to this
question is the class of map graphs (see Chapter 4).
For several problems in this chapter, Kloks et al. [50, 123, 107, 122] introduced a
reduction to the problem kernel on planar graphs. Because graphs excluding a singlecrossing graph are similar to planar graphs, in the sense of having a linear number
of edges and not having a clique of more than a constant size, we believe that one
might obtain similar results for these graphs.
As mentioned before, Theorem 3.20 holds for any class of graphs with treewidth
< 2. It is an open problem whether it is possible to generalize it to apply to any
class of graphs of treewidth < h for arbitrary fixed h. Moreover, there exists no
general method for designing O(6wn)-time algorithms for vertex-removal problems in
graphs with treewidth < w. If this becomes possible, then Theorem 3.23 will have
considerable algorithmic applications.
Finally, as a matter of practical importance, it would be interesting to obtain
a constant coefficient better than 9.55 for the treewidth of planar graphs having a
k-dominating set, which would lead to a direct improvement on our results.
87
88
Chapter 4
Fixed-Parameter Algorithms for
the (k,r)-Center in Planar Graphs
and Map Graphs
Clustering is a key tool for solving a variety of application problems such as data
mining, data compression, pattern recognition and classification, learning, and facility
location. Among the algorithmic problem formulations of clustering are k-means, kmedians, and k-center. In all of these problems, the goal is to partition n given points
into k clusters so that some objective function is minimized.
In this chapter, we concentrate on the (unweighted) (k, r)-center problem [26], in
which the goal is to choose k centers from the given set of n points so that every
point is within distance r from some center in the graph. In particular, the k-center
problem [102]of minimizing the maximum distance to a center is exactly (k, r)-center
when the goal is to minimize r subject to finding a feasible solution. In addition, the
r-domination problem [26, 100] of choosing the fewest vertices whose r-neighborhoods
cover the whole graph is exactly (k, r)-center when the goal is to minimize k subject
to finding a feasible solution.
A sample application of the (k, r)-center problem in the context of facility location
is the installation of emergency service facilities such as fire stations. Here we suppose
that we can afford to buy k fire stations to cover a city, and we require every building to
89
be within r city blocks from the nearest fire station to ensure a reasonable response
time. Given an algorithm for (k, r)-center, we can vary k and r to find the best
bicriterion solution according to the needs of the application. In this scenario, we
can afford high running time (e.g., several weeks of real time) if the resulting solution
builds fewer fire stations (which are extremely expensive) or has faster response time;
thus, we prefer fixed-parameter algorithms over approximation algorithms.
In this application, and many others, the graph is typically planar or nearly so.
Chen, Grigni, and Papadimitriou [58] have introduced a generalized notion of planarity which allows local nonplanarity. In this generalization, two countries of a map
are adjacent if they share at least one point, and the resulting graph of adjacencies is
called a map graph. (See Section 4.1 for a precise definition.) Planar graphs are the
special case of map graphs in which at most three countries intersect at a point.
Previous results. r-domination and k-center are NP-hard even for planar graphs.
For r-domination, the current best approximation (for general graphs) is a (log n+ 1)factor by phrasing the problem as an instance of set cover [26]. For k-center, there is a
2-approximation algorithm [102] which applies more generally to the case of weighted
graphs satisfying the triangle inequality.
Furthermore, no (2 -
)-approximation
algorithm exists for any e > 0 even for unweighted planar graphs of maximum degree 3
[137]. For geometric k-center in which the weights are given by Euclidean distance
in d-dimensional space, there is a PTAS whose running time is exponential in k [1].
Several relations between small r-domination sets for planar graphs and problems
about organizing routing schemes with compact structures is discussed in [100].
The (k, r)-center problem can be considered as a generalization of the well-known
dominating set problem. During the last two years in particular much attention has
been paid to constructing fixed-parameter algorithms with exponential speedup for
this problem. Alber et al. [2]were the first who demonstrated an algorithm checking
whether a planar graph has a dominating set of size < k in time 0(27° V n). This result
was the first non-trivial result for the parameterized version of an NP-hard problem
in which the exponent of the exponential term grows sublinearly in the parameter.
Recently, the running time of this algorithm was further improved to 0(227V'n) [116]
90
and 0(21513vk+n3+k4)
[93]. Fixed-parameter algorithms for solving many different
problems such as vertex cover, feedback vertex set, maximal clique transversal, and
edge-dominating set on planar and related graphs such as single-crossing-minor-free
graphs are considered in [74, 123](see also Chapter 3). Most of these problems have
reductions to the dominating set problem. Also, because all these problems are closed
under taking minors or contractions, all classes of graphs considered so far are minorclosed.
Our results. In this chapter, we focus on applying the tools of parameterized complexity, introduced by Downey and Fellows [83],to the (k, r)-center problem in planar
and map graphs. We view both k and r as parameters to the problem. We introduce
a new proof technique which allows us to extend known results on planar dominating
set in two different aspects.
First, we extend the exponential speed-up for a generalization of dominating set,
namely the (k, r)-center problem, on planar graphs. Specifically, the running time
of our algorithm is 0((2r +
1)6(2r+1)vf+12r+3/2n
+ n4 ), where n is the number of
vertices. Our proof technique is based on combinatorial bounds (Section 4.2) derived
from the Robertson, Seymour, and Thomas theorem about quickly excluding planar
graphs, and on a complicated dynamic program on graphs of bounded branchwidth
(Section 4.3). Second, we extend our fixed-parameter algorithm to map graphs which
is a class of graphs that is not minor-closed. In particular, the running time of the
corresponding algorithm is 0((2r + 1)6(4r+l)v+24r+3n+ n4 ).
Notice that the exponential component of the running times of our algorithms
depends only on the parameters, and is multiplicatively separated from the problem
size n.
Moreover, the contribution of k in the exponential part is sublinear.
In
particular, our algorithms have polynomial running time if k = O(log2 n) and r =
0(1), or if r = O(log n/ log log n) and k = 0(1). We stress the fact that we design
our dynamic-programming algorithms using branchwidth instead of treewidth because
this provides better running times.
Finally, in Section 4.5, we present several extensions of our results, including a
PTAS for the r-dominating set problem and a generalization to a broad class of graph
91
parameters.
4.1 Preliminary Results
In this section, we recall some definitions from Chapter 1 for which we present several
preliminary results in more detail.
(k, r)-center.
We say a graph G has a (k, r)-center or interchangeably has an r-
dominating set of size k if there exists a set S of centers (vertices) of size at most k
such that N&(S) = V(G). We denote by yr(G) the smallest k for which there exists
a (k, r)-center in the graph. One can easily observe that for any r the problem of
checking whether an input graph has a (k, r)-center, parameterized by k is W[2]-hard
by a reduction from dominating set. (See Downey and Fellows [83] for the definition
of the W Hierarchy.)
Map graphs. Let , be a sphere. A ,-plane graph G is a planar graph G drawn in
E. To simplify notation, we usually do not distinguish between a vertex of the graph
and the point of E used in the drawing to represent the vertex, or between an edge
and the open line segment representing it. We denote the set of regions (faces) in
the drawing of G by R(G). (Every region is an open set.) An edge e or a vertex v is
incident to a region r if e C f or v C , respectively. ( denotes the closure of r.)
For a E-plane graph G, a map M is a pair (G, q), where
: R(G)
{0, 1} is a
two-coloring of the regions. A region r E R(G) is called a nation if +(r) = 1 and a
lake otherwise.
Let N(M)
be the set of nations of a map M.
The graph F is defined on the
vertex set N(M), in which two vertices r1 , r 2 are adjacent precisely if rl n f2 contains
at least one edge of G. Because F is the subgraph of the dual graph G* of G, it is
planar. Chen, Grigni, and Papadimitriou [58] defined the following generalization of
planar graphs. A map graph GM of a map M is the graph on the vertex set N(M)
in which two vertices r1 , r 2 are adjacent in GM precisely if rFln
r2
contains at least
one vertex of G.
Recall from Chapter 1 that we denote by Gk the kth power of G, i.e., the graph
92
on the vertex set V(G) such that two vertices in Gk are adjacent precisely if the
distance in G between these vertices is at most k. Let G be a bipartite graph with a
bipartition U U W = V(G). The half square G2 [U] is the graph on the vertex set U
and two vertices are adjacent in G2 [U] precisely if the distance between these vertices
in G is 2.
Theorem 4.1 ([58]). A graph GM is a map graph if and only if it is the half-square
of some planar bipartitegraph H.
Here the graph H is called a witness for GM. Thus the question of finding a
(k, r)-center in a map graph GM is equivalent to finding in a witness H of GM a set
S C V(GM) of size k such that every vertex in V(GM) - S has distance < 2r in H
from some vertex of S.
The proof of Theorem 4.1 is constructive, i.e., given a map graph GM together
with its map M = (G, 0), one can construct a witness H for GM in time O(IV(GM)I+
IE(GM)).
One color class V(GM) of the bipartite graph H corresponds to the set
of nations of the map M.
Each vertex v of the second color class V(H) - V(GM)
corresponds to an intersection point of boundaries of some nations, and v is adjacent
(in H) to the vertices corresponding to the nations it belongs. What is important for
our proofs are the facts that
1. in such a witness, every vertex of V(H) - V(GM) is adjacent to a vertex of
V(GM), and
2. IV(H)I = O(IV(GM)I + IE(GM)I).
Thorup [162] provided a polynomial-time algorithm for constructing a map of a
given map graph in polynomial time. However, in Thorup's algorithm, the exponent
in the polynomial time bound is about 120 [57]. So from practical point of view there
is a big difference whether we are given a map in addition to the corresponding map
graph. Below we suppose that we are always given the map.
Branchwidth (see Section 2.4 for the definition and its relation to treewidth) is our
main tool in this chapter. All our proofs can be rewritten in terms of the related and
93
better-known parameter treewidth, and indeed treewidth would be easier to handle in
our dynamic program. However, branchwidth provides better combinatorial bounds
resulting in faster exponential speed-up of our algorithms.
The following deep result of Robertson, Seymour, and Thomas (Theorems (4.3)
in [145]and (6.3) in [151]) plays an important role in the results of this thesis.
Theorem 4.2 ([151]). Let k > 1 be an integer. Every planar graph with no (k x k)grid as a minor has branchwidth < 4k - 3.
4.2
Combinatorial Bounds
Lemma 4.3. Let p, k, r > 1 be integers and G be a planar graph having a (k, r)-center
and with a (p x p)-grid as a minor. Then k > (+12)2.
Proof. We set V = {1,...,p}
Let
x {1,...,p}.
F = (V,{((x, y), (x',y')) I Ix - x' + Iy - y'l = 1})
be a plane (p x p)-grid that is a minor of some plane embedding of G. W.l.o.g. we
assume that the external (infinite) face of this embedding of F is the one that is
incident to the vertices of the set Vext= {(x, y) x = 1 or x = p or y = 1 or y = p},
i.e., the vertices of F with degree < 4. We call the rest of the faces of F internal
faces. We set Vint = {(x, y) r + 1
x
p - r, r +1
y < p - r}, i.e., Vnt is the set
of all vertices of F within distance > r from all vertices in Vext. Notice that F[Vint]
is a sub-grid of F and IVnt = (p - 2r)2 . Given any pair of vertices (x, y), (x', y') E V
we define 6((x, y), (x', y')) = max{lx - x'l, Y-
'l
We also define dF((x, y), (x', y')) to be the distance between any pair of vertices
(x, y) and (x', y') in F. Finally we define J to be the graph occurring from F by
adding in it the edges of the following sets:
{((x,y), (x + 1, + 1) I 1 < x
94
- , 1 <y
p - 1)}
((x,y+ ),(x+ 1,y) I 1 x<p-1,
1 <yp-1)}
(In other word we add all edges connecting pairs of non-adjacent vertices incident
to its internal faces). It is easy to verify that V(x, y), (x', y') E V
((x, y), (x', y')) =
d ((x, y), (x', y')). This implies the following.
If R is a subgraph of J, then V(x, y), (x', y') E V 6((x, y), (x', y')) < dR((x, y), (x', y')}4.1)
For any (x, y) E V we define Br((x, y)) = ((a, b) E V I 6((x, y), (a, b)) < r} and we
observe the following:
V(,y)ev IV(B((x,y)))
< (2r + 1)2 .
(4.2)
Consider now the sequence of edge contractions/removals that transform G to F. If
we apply on G only the contractions of this sequence we end up with a planar graph
H that can obtained by the (p x p)-grid F after adding edges to non-consecutive
vertices of its faces. This makes it possible to partition the additional edges of H
into two sets: a set denoted by E1 whose edges connect non-adjacent vertices of some
square face of F and another set E 2 whose edges connect pairs of vertices in Vext.
We denote by R the graph obtained by F if we add the edges of E1 in F. As R is a
subgraph of J, (4.1) implies that
V(x,y)EV N ((x,
y)) C B((x,
(4.3)
))
We also claim that
V(X,Y)eV
NI((X,
y)) C B,((X, y)) U (V
-
Vint )
(4.4)
To prove (4.4) we notice first that if we replace H by R in it then the resulting relation
follows from (4.3). It remains to prove that the consecutive addition of edges of E2 in
R does not introduce in NR((x, y)) any vertex of Vint. Indeed, this is correct because
any vertex in Vextis in distance > r from any vertex in Vint. Notice now that (4.4)
95
implies that V(_,y)ev NT((x, Y)) fn vntC Br((x,y )) n
Vint
and using (4.2) we conclude
that
V(X,y)EV
N((x,y))
n Vint < (2r + 1)2
(4.5)
Let S be a (k', r)-center in the graph H. Applying (4.5) on S we have that the
r-neighborhood of any vertex in S contains at most (2r + 1)2 vertices from Vit.
Moreover, any vertex in Vi,t should belong to the r-neighborhood of some vertex in
S. Thus k' > IintI/(2r + 1)2 = (p - 2r) 2/(2r + 1)2 and therefore k' > (P-)2
Clearly, the conditions that G has an r-dominating set of size k and H __ G imply
that H has an r-dominating set of size k' < k. (But this is not true for H
G.) As
H is a contraction of G and G has a (k, r)-center, we have that k > k' > (P:)2
and
O
lemma follows.
We are ready to prove the main combinatorial result of this chapter:
Theorem 4.4. For any planar graph G having a (k,r)-center, bw(G) < 4(2r +
1)
+ 8r+ 1.
Proof. Supposethat bw(G) > p = 4(2r + 1)k + 8r + - 3 for some , 0 <
for which p + 3 _
< 4,
(mod 4). By Theorem 4.2, G contains a (p x p)-grid as a minor
where p = (2r + 1)V/ + 2r + . By Lemma 4.3, k > (:-2r)2 =
implies that JV > V- + +,
a contradiction.
((2r+1l)vk+4)2
which
O
Notice that the branchwidth of a map graph is unbounded in terms of k and r.
For example, a clique of size n is a map graph and has a (1, 1)-center and branchwidth
> 2/3n.
Theorem 4.5. For any map graph GM having a (k, r)-center and its witness H,
bw(H) < 4(4r + 3)V/ + 16r+ 9.
Proof. The question of finding a (k, r)-center in a map graph GM is equivalent to
finding in a witness H of GM a set S C V(GM) of size k such that every vertex
V(GM) - S is at distance < 2r in H from some vertex of S. By the construction
96
of the witness graph, every vertex of V(H) - V(GM) is adjacent to some vertex of
V(GM). Thus H has a (k, 2r + 1)-center and by Theorem 4.4 the proof follows.
4.3
(k,r)-Centers in Graphs of Bounded Branchwidth
In this section, we present a dynamic-programming approach to solve the (k, r)-center
problem on graphs of bounded branchwidth. It is easy to prove that, for a fixed r, the
problem is in MSOL (monadic second-order logic) and thus can be solved in linear
time on graphs of bounded treewidth (branchwidth). However, for r part of the input,
the situation is more difficult. Additionally, we are interested in not just a linear-time
algorithm but in an algorithm with running time f(k, r)n.
It is worth mentioning that our algorithm requires more than a simple extension of
Alber et al.'s algorithm for dominating set in graphs of bounded treewidth [2], which
corresponds to the case r = 1. In fact, finding a (k, r)-center is similar to finding
homomorphic subgraphs, which has been solved only for special classes of graphs and
even then only via complicated dynamic programs [105]. The main difficulty is that
the path v = v, Vl, v2,
...
,
V<r =
c from a vertex v to its assigned center c may
wander up and down the branch decomposition repeatedly, so that c and v may be in
radically different 'cuts' induced by the branch decomposition. All we can guarantee
is that the next vertex v1 along the path from v to c is somewhere in a common 'cut'
with v, and that vertex vl and v2 are in a common 'cut', etc. In this way, we must
propagate information through the vi's about the remote location of c.
Let (T', ) be a branch decomposition of a graph G with m edges and let w':
E(T')
2V(G)
be the order function of (T', T). We choose an edge {x, y} in T', put
a new vertex v of degree 2 on this edge, and make v adjacent to a new vertex r.
By choosing r as a root in the new tree T = T' U v, r}, we turn T into a rooted
tree. For every edge of f E E(T)n
E(T'), we put w(f) = w'(f).
w({x, v}) = w({v, y}) = w'({x, y}) and w({r, v}) = 0.
97
Also we put
For an edge f of T we define Ef (Vf) as the set of edges (vertices) that are
"below" f, i.e., the set of all edges (vertices) g such that every path containing g
and v, r} in T contains f. With such a notation, E(T) = Ev,r} and V(T) = V{v,r}.
Every edge f of T that is not incident to a leaf has two children that are the edges
of Ef incident to f.
We denote by Gf the subgraph of G induced by the vertices
incident to edges from the following set
{T-l(X)
Ix
E Vf A (x is a leaf of T')).
The subproblems in our dynamic program are defined by a coloring of the vertices
in w(f) for every edge f of T. Each vertex will be assigned one of 2r + 1 colors
{O, T1, 2,...,Tr,
1,12,...,
r}.
The meaning of the color of a vertex v is as follows:
* 0 means that the vertex v is a chosen center.
* i means that vertex v has distance exactly i to the closest center c. Moreover,
there is a neighbor u E V(Gf) of v that is at distance exactly i - 1 to the center
c. We say that neighbor u resolves vertex v.
* i means that vertex v has distance exactly i to the closest center c. However,
there is no neighbor of v in V(Gf) resolving v. Thus we are guessing that any
vertex resolving v is somewhere in V(G) - V(Gf).
Intuitively, the vertices colored by i have already been resolved (though the vertex
that resolves it may not itself be resolved), whereas the vertices colored by Ti still
need to be assigned vertices that are closer to the center.
We use the notation
i to denote a color of either Ti or i. Also we use 10 = 0.
For an edge f of T, a coloring of the vertices in w(f) is called locally valid if the
following property holds: for any two adjacent vertices v and w in w(f), if v is colored
Ii and w is colored
Ij,
then i - jl < 1. (If the distance from some vertex v to the
98
closest center is i, then for every neighbor u of v the distance from u to the closest
center can not be less than i - 1 or more than i + 1.)
For each locally valid coloring c of w(f), f E E(T), we define Af(c) as the size of
the "minimum (k, r)-center restricted to Gf and coloring c". More precisely, Af(c) is
the minimum cardinality of a set Df(c) C V(Gf) such that
* For every vertex v E w(f),
- c(v) = 0 if and only if v E Df(c), and
- if c(v) =li, i > 1, then v
colored by
Ij, j
Df(c) and either there is a vertex u E w(f)
< i, at distance i -j
from v in Gf, or there is a path P of
length i in Gf connecting v with some vertex of Df (c) such that no inner
vertex of P is in w(f).
* Every vertex v E V(Gf) - w(f) whose closest center is at distance i, either is
at distance i in Gf from some center in Df(c), or is at distance j, j < i, in Gf
from a vertex u E w(f) colored I(i - j).
We put Af(c) = +oo if there is no such a set Df(c), or if c is not a locally valid
coloring. Because w({r, v}) = 0 and G{r,v}= G, we have that A{r,v}(c)is the smallest
size of an r-dominating
set in G.
We start computations of the functions Af from leaves of T. Let x be a leaf of
T and let f be the edge of T incident with x. Then Gf is the edge {u, v} of G
correspondingto x and either V(Gf) = w(f), or the vertex u = V(Gf) - w(f) is the
vertex of degree 1 in G. If V(Gf) = w(f), we consider all locally valid colorings c of
w(f) such that if a vertex v E w(f) is colored by
i for i > 0 then u is colored by
Ii - 1. Let us note that there is always an optimal solution containing no centers
in vertices of degree 1. Thus in the case v = V(Gf) - w(f) we color v in one of the
colors from the set {0, T1, T2, . . ., Tr - 1}. For each such coloring c we define Af(c)
to be the number of vertices colored 0 in w(f).
Otherwise, Af(c) is +oo, meaning
that this coloring c is infeasible. The brute-force algorithm takes O(rm) time for this
step.
99
Let f be a non-leaf edge of T and let fl,
w(f2), X2 = w(f)-W(fi),
X3
=
f2
be the children of f. Define X1 = w(f)-
W(f)nw(fi)nw(f
2
), and X4 = ((fi)UW(f
2 ))-(f).
Notice that
(f) = X1 U X 2 U X3 .
By the definition of
(4.6)
, it is impossible that a vertex belongs to exactly one of
w(f), W(fi), (f 2 ). Therefore, condition u E X 4 implies that u E w(f 1 ) n W(f2 ) and
we conclude that
w(fi) = X U X3 U X 4 ,
(4.7)
W(f 2 ) = X 2 U X 3 U X4 .
(4.8)
and
We say that a coloring c E {O,T1, T2, ... , Tr, 11, 12,...,
r}Iw(f) of w(f) is formed
from a coloring c1 of w(fl) and a coloring c2 of w(f 2 ) if
1. For every u e X1 , c(u) = cl(u);
2. For every u E X 2, c(u) = c2(u);
3. For every u E X 3,
(a) If c(u) =Ti, 1 < i < r, then c(u) = cl(u) =
2 (u).
Intuitively, because
vertex u is unresolved in w(f), this vertex is also unresolved in w(fi) and
in w(f2).
(b) If c(u) = 0 then c(u) = c2 (u) = 0.
(c) If c(u) =li, 1 < i < r, then c1(u), c2 (u) E {i,
i} and cl(u) y~c2 (u). We
avoid the case when both cl and c2 are colored by Li because it is sufficient
to have the vertex u resolved in at least one coloring. This observation
helps to decrease the number of colorings forming a coloring c. (Similar
100
arguments using a so-called "monotonicity property" are made by Alber et
al. [2] for computing the minimum dominating set on graphs of bounded
treewidth.)
4. For every u E X4,
(a) either c(u) = C2 (U) =
0
(in this case we say that u is formed by 0 colors),
(b) or Cl(u), c2 (u) E {(i, Ti} and cl(u)
=I
c2 (u), 1
i < r (in this case we say
that u is formed by (li, Ti} colors).
This property says that every vertex u of w(fi) and w(f 2 ) that does not appear
in w(f) (and hence does not appear further) should finally either be a center (if
both colors of u in cl and c2 were 0), or should be resolved by some vertex in
V(Gf) (if one of the colors cl(u), c2 (u) is i and one Ti). Again, we avoid the
case of Pi in both cl and c2.
Notice that every coloring of w(f) is formed from some colorings of w(fi) and
w(f 2 ). Moreover, if D (c) is the restriction to Gf of some (k, r)-center and such a restriction corresponds to a coloring c of w(f) then Df(c) is the union of the restrictions
Df (cl), Df2 (c2) to Gfh, Gf2 of two (k, r)-centers where these restrictions correspond
to some colorings cl, c2 of w(fi) and w(f 2 ) that form the coloring c.
We compute the values of the corresponding functions in a bottom-up fashion.
The main observation here is that if fi and
f2
are the children of f, then the vertex
sets w(fl) w(f 2 ) "separate" subgraphs Gf1 and Gf2, so the value Af(c) can be obtained
from the information on colorings of w(fi) and w(f 2 ).
More precisely, let c be a coloring of w (f) formed by colorings cl and c2 of fi and
f2.
Let #o(X 3, c) be the number of vertices in X 3 colored by color 0 in coloring c and
and let #o(X 4 , c) be the number of vertices in X4 formed by 0 colors. For a coloring
c we assign
Af(c) = min{Af, (cl) + Af2 (c2 ) - #o(X 3 , c1) - #o(X4 , c1)
1, C2
form c}.
(Every 0 from X3 and X4 is counted in Af,(cl) + Af 2 (c2 ) twice and X 3 n X
101
(4.9)
4 =
0.)
The time to compute the minimum in (4.9) is given by
O(Z
{{cic 2 } Ci,C2 form c} )
Let xi = Xi, 1 < i < 4. For a coloring c let z 3 be the number of vertices colored
by I colors in X3 . Also we denote by z 4 the number of vertices in X4 formed by
4.
2 3+Z
The number
of colorings of w(f) such that exactly z 3 vertices of X3 are colored by
colors and
{ti, ti} colors, 1 < i < r. Thus the number of pairs forming c is
such that exactly
4
vertices of X4 are formed by {l, T} colors is
(2r+ 1)x1(2r + 1)X2
(r
+
()
1)3-z3
()
ZZ4
rZ4
Thus the number of operations needed to estimate (4.9) for all possible colorings of
w(f) is
E E 2P+(2r + 1)x1+x2(r
+ 1)3(- P
) rP("
)rQ
= (2r + 1)1+2+4(3r + 1)X3
p=O q=O
Let e be the branchwidth of G. The sets Xi, 1 < i < 4, are pairwise disjoint and
by (4.6)-(4.8),
xl +X 2 +X3
< f
Xl +X3+X4
<
e
X2+X3+X4
<
e.
(4.10)
The maximum value of the linear function log3r+l(2r + 1) (X1 + x2 + x 4 ) + X 3
subject to the constraints in (4.10) is 3109
3r+1(2r+l1)f(This is because the value of the
corresponding LP achieves maximum at X1 =
2 = X4 = 0.5£, x 3 = 0.) Thus one can
evaluate (4.9) in time
+ 1)z 3 < - (3r - ) 3 Og3r+(2r+l )
(2r +
1)zl+z2+4(3r
(2r + 1)x1+x2+x4(3r+ 1)X3• (3r ± 1)
2
102
(2
-(2r
+
1)
It is easy to check that the number of edges in T is O(m) and the time needed to
evaluate A{r,v}(c) is 0((2r + 1)2 m). Moreover, it is easy to modify the algorithm to
obtain an optimal choice of centers by bookkeeping the colorings assigned to each set
w(f).
Summarizing, we obtain the following theorem:
Theorem 4.6. For a graph G on m edges and with a given branch decomposition of
width < , and integers k, r, the existence of a (k, r)-center in G can be checked in
0((2r + 1)2tm)
time and, in case of a positive answer, constructs a (k, r)-center of
G in the same time.
Similar result can be obtained for map graphs.
Theorem 4.7. Let H be a witness of a map graph GM on n vertices and let k, r
be integers. If a branch-decompositionof width < £ of H is given, the existence of a
(k, r)-center in GM can be checkedin 0((2r + 1)e n) time and, in case of a positive
answer, constructs a (k, r)-center of G in the same time.
Proof. We give a sketch of the proof here. H is bipartite graph with a bipartition
(V(GM),V(H)-
V(GM)).
There is a (k,r)-center
in GM if and only if H has a
set S C V(GM) of size k such that every vertex V(GM) - S is at distance < 2r in
H from some vertex of S. We check whether such a set S exists in H by applying
arguments similar the proof of Theorem 4.6. The main differences in the proof are
the following. Now we color vertices of the graph H by I i, 0 < i < 2r, where i is
even. Thus we are using 2r + 1 numbers. Because we are not interested whether the
vertices of V(H) - V(GM) are dominated or not, for vertices of V(H) - V(GM) we
keep the same number as for a vertex of V(GM) resolving this vertex. For a vertex
in V(GM) we assign a number i if there is a resolving vertex from V(H) - V(GM)
colored I (i - 2). Also we change the definition of locally valid colorings: for any
two adjacent vertices v and w in w(f), if v is colored
i and w is colored j, then
i- il < 2.
Finally, H is planar, so IE(H)J= O(IV(H)I) =
103
O(n).
°]
4.4 Algorithms for the (k, r)-Center Problem
For a planar graph G and integers k, r, we solve (k, r)-center problem on planar graphs
in three steps.
Step 1: We check whether the branchwidth of G is at most 4(2r + 1)vk + 8r + 1.
2 ) time according to the algorithm due to
This step requires O((IV(G)I + IE(G)I)
Seymour & Thomas (algorithm (7.3) of Section 7 of [155] - for an implementation,
see the results of Hicks [113]). If the answer is negative then we report that G has no
any (k, r)-center and stop. (The correctness of this step is verified by Theorem 4.4.)
Otherwise go to the next step.
Step 2: Compute an optimal branch-decomposition of a graph G. This can be done
by the algorithm (9.1) in the Section 9 of [1551which requires O((IV(G)I + E(G)1)4 )
steps.
Step 3: Compute, if it exists, a (k, r)-center of G using the dynamic-programming
algorithm of Section 4.3.
It is crucial that, for practical applications, there are no large hidden constants in
the running time of the algorithms in Steps 1 and 2 above. Because for planar graphs
IE(G)I = O(IV(G)[), we conclude with the following theorem:
Theorem 4.8. There exists an algorithm finding, if it exists, a (k,r)-center of a
planargraphin 0((2r+
1)6(2r+l)vr/+12r+3/2n+ n4 )
time.
Similar arguments can be applied to solve the (k, r)-center problem on map graphs.
Let GM be a map graph. To check whether GM has a (k,r)-center, we compute
optimal branchwidth of its witness H. By Theorem 4.5, if bw(H) > 4(4r + 3)VT +
16r + 9, then GM has no (k, r)-center. If bw(H) < 4(4r + 3)vk + 16r + 9, then by
Theorem 4.7 we obtain the following result:
Theorem 4.9. There exists an algorithmfinding, if it exists, a (k, r)-center of a map
graphin 0((2r +
1)6(4r+l)v'k+24r+13.5n
+ n4 ) time.
104
By a straightforward modification to the dynamic program, we obtain the same
results for the vertex-weighted (k, r)-center problem, in which the vertices have real
weights and the goal is to find a (k, r)-center of minimum total weight.
4.5 Concluding Remarks
In this chapter, we presented fixed-parameter algorithms with exponential speed-up
for the (k, r)-center problem on planar graphs and map graphs. Our methods for
(k, r)-center can also be applied to algorithms on more general graph classes like constant powers of planar graphs, which are not minor-closed family of graphs. Extending
these results to other non-minor-closed families of graphs would be instructive.
Surprisingly, the algorithm described in Section 4.4 does not use special properties
of (k, r)-center problem at all. The only properties the algorithm really needs are the
combinatorial bound used in Step 1 and the fact that the problem can be solved
on graphs of bounded branchwidth (Step 3). The proof of combinatorial bound
(Theorem 4.4) is based on excluded planar graphs theorem of Robertson, Seymour &
Thomas and the following two facts used in the proof of Lemma 4.3.
Fact 1: (k, r)-center problem is closed under edge contraction operation
Fact 2: For any partially triangulated grid (a partially triangulated (p x p)-grid is any
graph obtained by adding non-crossing edges between pairs of nonconsecutive
vertices on a common face of a planar embedding of an (p x p)-grid) having a
(k, r)-center, k is at least (P1:r)2
The above observations, summarized in the next theorem, provide a general and
versatile approach for many parameterized problems.
Theorem 4.10. Let p be a function mapping graphs to non-negativeintegers such
that the following conditions are satisfied:
(1) There exists an algorithm checking whether p(G) < w in f(bw(G))n
105
0 (1
) steps.
Figure 4-1: A partially triangulated (12 x 12)-grid.
(2) For any k > 0, the class of graphs where p(G) < k is closed under taking of
contractions.
(3) For any partially triangulated (p x p)-grid R, p(R) =
Then there exists an algorithm checking whether p(G)
(p2).
k on planar graphs in
O(f(v))nO(1) steps.
For a wide source of parameters satisfying condition (1) we refer to the theory of
Courcelle [61] (see also [15]). Apart from (k, r)-center and dominating set, examples
of parameters satisfying conditions (2) and (3) are vertex cover, feedback vertex set,
minimum maximal matching, edge dominating set and many others. For parameters
where f(bw(G))
= 20(bw(G)),this result is a strong generalization of Alber et al.'s
approach which requires that the problem of checking whether p(G) < k should
satisfy the "layerwise separation property" [6]. Moreover, the algorithms involved are
expected to have better constants in their exponential part comparatively to the ones
appearing in [6]. More generally, it seems that our approach should extend other graph
algorithms (not just dominating-set-type problems) to apply to the rth power and/or
half-square of a graph (and hence in particular map graphs). It would be interesting
to explore to which other problems our approach can be applied. Also, obtaining
"fast" algorithms for problems like feedback vertex set or vertex cover on constant
106
powers of graphs of bounded branchwidth (treewidth), as we did for dominating set,
would be interesting.
In addition, there are several interesting variations on the (k, r)-center problem.
In multiplicity-m (k, r)-center, the k centers must satisfy the additional constraint
that every vertex is within distance r of at least m centers. In f-fault-tolerant (k, r)center [26], every non-center vertex must have f vertex-disjoint paths of length at
most r to centers.
(For this problem with r = oo, [26] gives a polynomial-time
O(f log IVI)-approximation algorithm for k.) In L-capacitated (k, r)-center [26], each
of the k centers can satisfy only L "customers", essentially forcing the assignment of
vertices to centers to be load-balanced. (For this problem, [26] gives a polynomialtime O(log Vl)-approximation algorithm for r.) In connected (k, r)-center [157],the k
chosen centers must form a connected subgraph. In all these problems, the main challenge is to design the dynamic program on graphs of bounded treewidth/branchwidth.
We believe that our approach can be used as the main guideline in this direction.
Map graphs can be seen as contact graphs of disc homeomorphs. A question is
whether our results can be extended for another geometric classes of graphs. An
interesting candidate is the class of unit-disk graphs. The current best algorithms
V )
for finding a vertex cover or a dominating set of size k on these graphs have nO(1
running time [7].
Using our results we can also easily obtain a PTAS for r-dominating set on planar and map graphs. These results are similar to the approximation algorithms for
independent set on map graphs by Chen [57]. We combine Theorems 4.6 and 4.7
with the approaches of Eppstein [87] and Grohe [103] (which in turn are based on
the classic Baker's approach [23]), and adapt these approaches to branch decompositions instead of tree decompositions. We obtain a (1 + 2r/p)-approximation algo-
rithm for r-domination in planar graphs with running time O(p(2r + 1)3 (P+2r)m), and
for map graphs we obtain a (1 + 4r/p)-approximation algorithm with running time
O(p(4r + 3) 3 (p+4r)m).
107
108
Chapter 5
Subexponential Parameterized
Algorithms on Bounded-Genus
Graphs and H-Minor-Free Graphs
Algorithms for H-minor-free graphs for a fixed graph H have been studied extensively;
see e.g. [51, 104, 53, 120, 138]. In particular,
it is generally believed that several
algorithms for planar graphs can be generalized to H-minor-free graphs for any fixed
H [104, 120, 138]. H-minor-free graphs are very general. The deep Graph-Minor
Theorem of Robertson and Seymour shows that any graph class that is closed under
minors is characterized by excluding a finite set of minors. In particular, any graph
class that is closed under minors (other than the class of all graphs) excludes at least
one minor H.
In this chapter, we introduce a framework for extending algorithms for planar
graphs to apply to H-minor-free graphs for any fixed H. In particular, we design
subexponential fixed-parameter algorithms for dominating set, vertex cover, and set
cover (viewed as one-sided domination in a bipartite graph) for H-minor-free graphs.
Our framework consists of three components, as described below. We believe that
many of these components can be applied to other problems and conjectures as well
(see e.g., Chapter 8 for another application of this framework).
First we extend the algorithm for planar graphs to bounded-genus graphs. Roughly
109
speaking, we study the structure of the solution to the problem in k x k grids, which
form a representative substructure in both planar graphs and bounded-genus graphs,
and capture the main difficulty of the problem for these graphs. Then using Robertson and Seymour's graph-minor theory, we repeatedly remove handles to reduce the
bounded-genus graph down to a planar graph, which is essentially a grid.
Second we extend the algorithm to almost-embeddable graphs which can be drawn
in a bounded-genus surface except for a bounded number of "local areas of nonplanarity", called vortices, and for a bounded number of "apex" vertices, which can
have any number of incident edges that are not properly embedded (see Section 1.3
for precise definitions). Because each vortex has bounded pathwidth, the number of
vortices is bounded, and the number of apices is bounded, we are able to extend our
approach to solve almost-embeddable graphs using our solution to bounded-genus
graphs.
Third we apply a deep theorem of Robertson and Seymour which characterizes
H-minor-free graphs as a tree structure of pieces, where each piece is an almostembeddable graph. Using dynamic programming on such tree structures, analogous
to algorithms for graphs of bounded treewidth, we are able to combine the pieces
and solve the problem for H-minor-free graphs. Note that the standard boundedtreewidth methods do not suffice for general H-minor-free graphs, unlike e.g. boundedgenus graphs, because their treewidth can be arbitrarily large with respect to the
parameter [62] (see Chapter 7). Our contribution is to overcome this barrier algorithmically using a two-level dynamic program in a more general tree structure called a
"clique-sum decomposition".
The first step of this procedure, for bounded-genus graphs, applies to a broad
class of problems called "bidimensional problems". Roughly speaking, a parameterized problem is bidimensional if the parameter is large (linear) in a grid and closed
under contractions. Examples of bidimensional problems include vertex cover, feedback vertex set, minimum maximal matching, dominating set, edge dominating set,
clique-transversal set, and set cover. We obtain subexponential fixed-parameter algorithms for all of these problems in bounded-genus graphs. As a special case, this
110
generalization settles an open problem about dominating set posed by Ellis, Fan,
and Fellows [84]. Along the way, we establish an upper bound on the treewidth
(or branchwidth) of a bounded-genus graph that excludes some planar graph H as
a minor (See Section 2.4 for the exact definition of branchwidth and its relation to
treewidth). This bound depends linearly on the size IV(H)l of the excluded graph H
and the genus g(G) of the graph G, and applies and extends the graph-minors work
of Robertson and Seymour.
This chapter is organized as follows. Section 5.1 is devoted to graphs on surfaces.
We construct a general framework for obtaining subexponential parameterized algorithms on graphs of bounded genus. First we introduce the concept of bidimensional
problem, and then prove that every bidimensional problem has a subexponential parameterized algorithm on graphs of bounded genus. The proof techniques used in
this section are very indirect and are based on deep theorems from Robertson and
Seymour's Graph Minors XI and XII. As a byproduct of our results we obtain a
generalization of Quickly Excluding a Planar Graph Theorem for graphs of bounded
genus. In Section 5.3 we make a further step by developing subexponential algorithms
for graphs containing no fixed graph H as a minor. The proof of this result is based
on combinatorial bounds from the previous section, a deep structural theorem from
Graph Minors XIV, and complicated dynamic programming. Finally, in Section 5.4,
we present several extensions of our results and some open problems.
5.1
Graphs on Surfaces
5.1.1
Preliminaries
In this subsection we describe some of the machinery developed in the Graph Minors
series that we use in our proofs.
A surface E is a compact 2-manifold without boundary. A line in E is a subset
homeomorphic to [0,1]. An O-arc is a subset of Z homeomorphic to a circle. A subset
of E is an open disc if it is homeomorphic to {(x, y) x 2 + y2 < 1}, and it is a closed
111
disc if it is homeomorphic to {(x, y) I x 2 + y2 < 1}.
A 2-cell embedding of a graph G in a surface P is a drawing of the vertices as
points in , and the edges as lines in Y such that every region (face) bounded by
edges is an open disc. To simplify notation, we do not distinguish between a vertex of
G and the point of E used in the drawing to represent the vertex, or between an edge
and the line representing it. We also consider G as the union of points corresponding
to its vertices and edges. Also, a subgraph H of G can be seen as a graph H where
H C G. A region of G is a connected component of Y - E(G) - V(G).
(Every
region is an open disc.) We use the notation V(G), E(G), and R(G) for the set of
the vertices, edges, and regions of G.
If A C E, then A denotes the closure of A, and the boundary of A is bd(A) =
A NY - A. A vertex or an edge x is incident to a region r if x C bd(r).
A subset of P meeting the drawing only at vertices of G is called G-normal. If an
O-arc is G-normal, then we call it a noose. The length of a noose is the number of its
vertices. We say that a disc D is bounded by a noose N if N = bd(D). A graph G
2-cell embedded in a connected surface E is
0-representative
if every noose of length
less than 0 is contractable (null-homotopic in E).
Tangles were introduced by Robertson and Seymour in [145]. A separation of a
graph G is a pair (A, B) of subgraphs with A U B = G and E(A n B) = 0, and its
order is IV(A n B)I. A tangle of order 0 > 1 is a set T of separations of G, each of
order less than 0, such that
1. for every separation (A, B) of G of order less than 0, T contains one of (A, B)
and (B, A);
2. if (Al, B),
(A 2 , B 2 ), (A3, B3) E T, then Al U A2 U A 3 =#G; and
3. if (A, B) E T, then V(A)
V(G).
Let G be a graph 2-cell embedded in a connected surface Y. A tangle T of order
0 is respectful if, for every noose N in E of length less than 0, there is a closed disc
A C E with bd(A) = N such that the separation (G n A, G n E - A) E T.
112
Our proofs are based on the following results from the Graph Minors series of
papers by Robertson and Seymour.
Theorem 5.1. [145, (4.3)] Let G be a graph with at least one edge. Then there is a
tangle in G of order 0 if and only if G has branchwidthat least 0.
Theorem
5.2. [146, (4.1)] Let ) be a connected surface, not homeomorphic to a
sphere; let 0 > 1; and let G be a -representative graph 2-cell embedded in E. Then
there is a unique respectful tangle in G of order 0.
Our proofs also use the notion of the radial graph. Informally, the radial graph
of a graph G 2-cell embedded in E is the bipartite graph RG obtained by selecting
a point in every region r of G and connecting it via an edge to every vertex of G
incident to that region. However, a region may be incident to the same vertex "more
than once", so we need a more formal definition. Precisely, RG is a radial graph of a
graph G 2-cell embedded in E if
1. E(G) n E(RG) = V(G) C V(RG);
2. each region r E R(G) contains a unique vertex v, E V(RG);
3. RG is bipartite with a bipartition (V(G), {vr: r E R(G)});
4. if e, f are edges of RG with the same ends v E V(G), v, E V(RG), then e U f
does not bound a closed disc in r U {v}; and
5. RG is maximal subject to Conditions 1-4.
5.1.2
Bounding the Representativity
Define the (r x r)-grid to be the graph on r 2 vertices {(x, y) I 1 < x, y < r} with edges
between vertices differing by -1 in exactly one coordinate. A partially triangulated
(r x r)-grid is any planar supergraph of the (r x r)-grid.
Lemma 5.3. Let G be a graph 2-cell embedded in a surface E, not homeomorphic to
a sphere, of representativity at least 0. Then G contains as a contraction a partially
triangulated(/4 x /4)-grid.
113
Proof. By Theorem 5.2, G has a respectful tangle of order 0. Let A(RG) be the set of
vertices, edges, and regions (collectively, atoms) in the radial graph RG. According to
[146, Section 9] (see also [147]), the existence of a respectful tangle makes it possible
to define a metric d on A(RG) as follows:
1. If a = b, then d(a, b) = 0.
2. If a
b, and a and b are interior to a contractible closed walk in the radial graph
of length less than 20, then d(a, b) is half the minimum length of such a walk.
(Here by interior we mean the direction in which the walk can be contracted.)
3. Otherwise,
d(a, b) = 0.
Assume for simplicity that 0 is even and that
For 0
> 4. Let c be any vertex in G.
i < 0/2, define Z2 i to be the union of all atoms of distance at most 2i
from c (where distance is measured according to the metric d). (Notice that, in radial
graphs, all closed walks have even length.) By [146, (8.10)], Z2i is a nonempty simply
connected set, for all i. (A subset of a surface is simply connected if it is connected
and has no noncontractible closed curves.) Thus, the boundary bd(Z2i) of each Z2i
is a closed walk in the radial graph.
We claim that the closed walks bd(Z 2 i ) and bd(Z2 i+2 ) are vertex-disjoint. Consider any atom a on bd(Z 2 i) and an adjacent atom b outside Z2 i. The distance
between a and b is 2 because there is a length-2 closed walk connecting them, doubling the edge (a,b). By Theorem 9.1 of [146], the metric d satisfies the triangle
inequality, and hence d(c, b) < d(c, a) + 2 = 2i + 2. In fact, this bound must hold
with equality, because b
Z2i. Therefore, every atom a on bd(Z2i) is surrounded on
the exterior of Z2 i by atoms at distance exactly 2i + 2 from c, so bd(Z 2 i) is strictly
enclosed by bd(Z 2 i+2).
Consider the "annulus" A = (Z 2 -2 - Zo) U bd(Z 2 0_2 ) U bd(Zo), which includes
the boundary bd(A) = bd(Z 2-
2)
U bd(Zo). We claim that there are at least 0/2
vertex-disjoint paths in the radial graph within A connecting vertices in bd(Ze) to
vertices in bd(Z 2 0_ 2 ). By Menger's Theorem, the contrary implies the existence of a
114
cut in A of size less than 0/2 separating the two sets, which implies the existence of
a cycle of length less than 0, but such a cycle must be contained in Zo.
Now we form a (/2 x 0/2)-grid in the radial graph. The row lines in the grid are
formed by taking, for each i = 8, 0 + 2, 0 + 4,..., 28 - 2, the unique simple cycle that
encloses c and that is a subset of the closed walk bd(Z 2i), The column lines in the
grid are formed by the
/2 vertex-disjoint paths found above. Therefore, we obtain
a subdivision of the (/2 x 0/2)-grid as a subgraph of the radial graph.
Finally, we transform this grid into a (/4
x 0/4)-grid in the original graph G.
Each grid edge in the radial graph corresponds in the original graph to a sequence
of faces surrounding the edge. We replace this grid edge by the upper half of each
face. In this way, each row line in the radial graph maps in the original graph to a
curve above this row line. Two adjacent mapped row lines may touch but cannot
properly cross, so row lines of distance 2 or more in the grid cannot overlap when
mapped to the original graph.
Thus, by discarding the odd-numbered row lines,
and similarly for the columns, we obtain a subdivision of the (/4 x /4)-grid in the
original graph. Because each Z2i was simply connected, the grid is embedded in a
simply connected subset of Z, so if we apply contractions without deletions, we obtain
a partially triangulated grid.
El
5.2 Bidimensional Parameters and Bounded-Genus
Graphs
In this section, we define a general framework of parameterized problems for which
subexponential algorithms with small constants can be obtained.
Our framework
is sufficiently broad that an algorithmic designer needs to check only two simple
properties of any desired parameter to determine the applicability and practicality of
our approach.
115
5.2.1
Definitions
Recall from Section 5.1.2 that a partially triangulated (r x r)-grid is any planar graph
obtained by adding edges between pairs of nonconsecutive vertices on a common face
of a planar embedding of an (r x r)-grid.
Definition 5.4. A parameter P is any function mapping graphs to nonnegative integers. The parameterized problem associated with P asks, for some fixed k, whether
P(G) < k for a given graph G.
Definition 5.5. A parameter P is minor bidimensional with density 6 if
1. contracting or deleting an edge in a graph G cannot increase P(G), and
2. for the (r x r)-grid R, P(R) = (6r)2 + o((6r)2 ).
A parameter P is called contraction bidimensional with density 6 if
1. contracting an edge in a graph G cannot increase P(G),
2. for any partially triangulated(r x r)-grid R, P(R) > (r) 2 + o((6r)2 ), and
3.
is the smallest real number for which this inequality holds.
In either case, P is called bidimensional. The density
of P is the minimum of
the two possible densities (when both definitions are applicable). We call the sublinear
function f(x) = o(x) in the bound on P(R) the residual function of P.
Notice that density assigns a positive real number, typically at most 1, to any
bidimensional parameter. Interestingly, this assignment defines a total order on all
such parameters.
5.2.2
Examples
Many parameters are bidimensional. Here we mention just a few. Examples of minorbidimensional parameters are
116
Vertex cover. A vertex cover of a graph G is a set C of vertices such that every
edge of G has at least one endpoint in C. The vertex-cover problem is to find a
minimum-size vertex cover in a given graph G. The corresponding parameter,
the size of a minimum vertex cover, is minor bidimensional with density
=
1/X/2. (Roughly half the vertices must be in any vertex cover of the grid, and
one color class in a vertex 2-coloring of the grid is a vertex cover.)
Feedback vertex set. A feedback vertex set of a graph G is a set U of vertices such
that every cycle of G passes through at least one vertex of U. The size of a
minimum feedback vertex size is a minor-bidimensional parameter with density
6 E [1/2, 1/X].
(6 > 1/2 because there are r2 /4 + o(r2 ) vertex-disjoint squares
in the (r x r)-grid, each of which must be broken; 6 < 1/v2 because it suffices
to remove one color class in a vertex 2-coloring of the grid.)
Minimum maximal matching.
A matching in a graph G is a set E' of edges with-
out common endpoints. A matching in G is maximal if it is contained by no
other matching in G. The size of a minimum maximal matching is a minorbidimensional parameter with density 6 E [1/V4, 1/X].
(6 > 1/N/8 because
any maximal matching must include at least one edge interior to any 3 x 4
subgrid, and there are r2 /8 + o(r2 ) interior-disjoint 3 x 4 subgrids;
< 1/vx/
because the number of edges in a matching is at most r2 /2.)
Examples of contraction-bidimensional parameters are
Dominating
set. A dominating set of a graph G is a set D of vertices of G such that
each of the vertices of V(G) - D is adjacent to at least one vertex of D. The
size of a minimum dominating set is a contraction-bidimensional parameter with
density
= 1/3. (6 > 1/3 because every vertex dominates at most 9 vertices;
6 < 1/3 because there is a triangulation of the (r x r)-grid with dominating set
of size r 2 /9 + o(r2 ).)
Edge dominating
set. An edge dominating set of a graph G is a set D of edges
of G such that every edge in E(G) - D shares at least one endpoint with
117
some edge in D. The size of a minimum edge domainting set is a contractionbidimensional parameter with density 6 = 1/vT.
(6 > 1/
because every
edge in a triangulated grid dominates at most 14 edges; 6 < 1/vi4
because
size-14 neighborhoods of a diagonal edge can be tiled to form a triangulated
(r x r)-grid requiring r 2 /14 + o(r2 ) dominating edges.)
Many of our results can be applied not only to bidimensional parameters but
also to parameters that are bounded by bidimensional parameters. For example, the
clique-transversal number of a graph G is the minimum number of vertices intersecting
every maximal clique of G. This parameter is not contraction-bidimensional because
an edge contraction may create a new maximal clique and cause the clique-transversal
number to increase. On the other hand, it is easy to see that this graph parameter
always exceeds the size of a minimum dominating set. In particular, this fact can be
used to obtain a parameter-treewidth bound for the clique-transversal number.
Finally, it is worth mentioning that the definition of bidimensional parameters
will be further extended for the general class of H-minor-free graphs in Chapter 7.
5.2.3
Subexponential Algorithms and Planar Graphs
Almost all known techniques for obtaining subexponential parameterized algorithms
on planar graphs are based on the following "bounded-treewidth approach" [2, 93,
116]:
(II) Prove that tw(G) < cP(G
for some constant c;
(I2) Compute or approximate the treewidth (or branchwidth) of G;
(I3) Decide whether P(G) < k as follows. If the treewidth is more than cvk, then
the answer to the decision problem is NO. If treewidth is at most cvk, then run a
standard dynamic program for graphs of bounded treewidth in 2 0 (tw(G))nO(l)
=
2°(V/)n°(1) time.
All previously known ways of obtaining the most important step (I1) use rather
complicated techniques based on separators.
118
Next we give some hints why bidi-
mensional parameters are important for the design of subexponential algorithms by
showing how step (Ii) can be performed for planar graphs.
For every bidimensional parameter P and (r x r)-grid R, IV(R)- = O(P(R)), by
Theorem 4.2, we have the following proposition.
Proposition 5.6. Let P be a bidimensionalparameter. Then for any planar graph
G, tw(G)= O(P(G)).
The class of bidimensional parameterized problems contains all parameters known
from the literature to have subexponential parameterized algorithms for planar graphs
[3, 2, 6, 50, 123, 107]. Recently, Cai et al. [48] defined a class of parameters, Planar TMIN 1, and proved that, for every planar graph G and parameter P in Planar TMIN 1, tw(G) = O(
(G)). Every problem in Planar TMIN 1 can be expressed
as a special type of dominating-set problem on bipartite graphs. (We refer to [48]
for definitions and further properties of Planar TMIN1 .) Using Proposition 5.6 it is
possible to prove a similar result, establishing the bound tw(G) = O(P(G)
for
most parameters P in Planar TMIN 1.
It is tempting to wonder whether every parameter admitting a 2°(V)n(1)-time
algorithm on planar graphs is bidimensional.
5.2.4
Parameter-Treewidth Bound for Bounded-Genus Graphs
To extend Proposition 5.6 to graphs of bounded genus, more work needs to be done.
If P is a bidimensional parameter with density 6 and residual function f, then we
define the normalization factor of P to be the minimum positive number
such that
(6r)2 < (6r)2 + f(dr) for any r > 1.
Lemma 5.7. Let P be a contraction (minor) bidimensionalparameter with density
6. Then P(G) < ( r)2 implies that G excludes the (r x r)-grid as a minor (and all
partial triangulations of the (r x r)-grid as contractions).
Proof. If P is minor bidimensional and H is the (r x r)-grid and H - G, then P(H) <
P(G). BecauseP(H) = (6r)2 + f(dr), we have that (r)
which contradicts the definition of 3.
119
2
> P(G) > (6r)2 + f(dr),
If P is contraction bidimensional and H is a partial triangulation of the (r x r)-
grid and H < G, then P(H) < P(G). Because P(H) = (6r)2 + f(Sr), we have that
('r)
> P(G) > (r)
2
+ f(Sr), which contradicts the definition of 3.
O
Let G be a graph and let v E V(G) be a vertex. Also suppose we have a partition
7Pv= (N1 , N 2 ) of the set of the neighbors of v. Define the splitting of G with respect
to v and P, to be the graph obtained from G by
1. removing v and its incident edges;
2. introducing two new vertices v1 and v 2; and
3. connecting vi with the vertices in Ni, for i = 1, 2.
If H is the result of consecutive application of several such operations to some graph G,
then we say that H is a splitting of G. If in addition the sequence of splittings never
splits a vertex that was the result of a previous splitting, then we say that H is a
fair splitting of G. The vertices v of G involved in the splittings that make up a fair
splitting are called affected vertices.
A parameter P is a-splittable if, for every graph G and for each vertex v E V(G),
the result G' of splitting G with respect to v satisfies P(G') < P(G) + a. Many natural graph problems are a-splittable for small constants a. Examples of 1-splittable
problems are dominating set, vertex cover, edge dominating set, independent set,
clique-transversal set, and feedback vertex set, among many others.
For the proof of our main result on properties of bidimensional parameters, we
need two technical lemmas used in induction on the genus.
It is convenient to work with Euler genus. The Euler genus eg(E) of a nonorientable surface E is equal to the nonorientable genus 9(S) (or the crosscap number).
The Euler genus eg(E) of an orientable surface E is 2g(S), where g(S) is the orientable
genus of E.
The following lemma is very useful in proofs by induction on the genus. The first
part of the lemma follows from [136, Lemma 4.2.4] (corresponding to a nonseparating
120
cycle) and the second part follows from [136, Proposition 4.2.1] (corresponding to a
surface-separating cycle).
Lemma 5.8. Let G be a connected graph 2-cell embedded in a surface E not homeomorphic to a sphere, and let N be a noncontractible noose on G. Then there is a fair
splitting G' of G affecting the set S = {vl,..., vp} of vertices of G met by N such
that one of the following holds:
1. G' can be 2-cell embedded in a surface with Euler genus strictly smaller than
eg(E); or
2. each connected component Gi of G' can be 2-cell embedded in a surface with
Euler genus strictly smaller than eg(Z) and is a contraction of some graphG*
obtainedfrom G after at most p splittings.
The following lemma is a direct consequence of the definition of branchwidth.
Lemma 5.9. Let G be a graph and let G' be the splitting of a vertex in G. Then
bw(G') < bw(G) + 1.
Theorem 5.10. Suppose that P is an a-splittable contraction bidimensionalparameter (a > 0) with density 6 > 0 and normalization factor
> 1. Then,
for any graph G 2-cell embeddedin a surface E of Euler genus eg(E), bw(G) <
4 (eg(E) + 1) +P(G)
1
+
8a( (eg(E) + 1))2.
Proof. We induct on the Euler genus of E.
In the base case that eg(E) = 0, Lemma 5.7 implies that, if P(G) < (r)
2,
then
G excludes the (r x r)-grid as a minor. This implication is precisely Lemma 5.7 in
the case that P is minor bidimensional. If P is contraction bidimensional, then the
implication follows because, if a planar graph G can be transformed to a graph H
(e.g., the (r x r)-grid) via a sequence of edge contractions and/or removals, then by
applying only the contractions in this sequence, we obtain a partial triangulation of H
as a contraction of G. Now by Theorem 4.2, if P(G) < (r)2,
If we set r =
then bw(G) < 4r - 6.
JP(G
+ 1, we have that bw(G) < 4LO P()J
a, 3, > 0, the induction base follows.
121
- 2. Because
split
S
S1
S2
Figure 5-1: Splitting a noose.
Suppose now that eg(E) > 1 and that the induction hypothesis holds for any
graph 2-cell embedded in a surface with Euler genus less than eg(E). Let G be a
graph embedded in E. We set k = P(G) and claim that the representativity of G
is at most 43lr
+lJ.
Lemma 5.7 implies that, if k < (r)
2,
then G excludes any
triangulation of the (r x r)-grid as a contraction. By the contrapositive of Lemma 5.3,
this implies that the representativity of G is less than 4r. If we set r = L vk +J + 1,
we have that the representativity of G is at most 4L3 v/kl + J. Let N be a minimum
size non-contractible noose N on E meeting p vertices of G where p < 4LOvl+
J.
By Lemma 5.8, there is a fair splitting along the vertices met by N such that either
Condition 1 or Condition 2 holds; see Figure 5-1. Let G' be the resulting graph
and let E' be a surface such that eg(E') < eg(E) - 1 and every component of G'
is 2-cell embedable in E'. We claim that, given either Condition 1 or Condition 2,
bw(G') < 4eg(E)/k + ap + 1 + 8a(3)2(eg())2.
Given Condition 1, we apply the induction hypothesis to G' and get that bw(G') <
40(eg(E') + 1)V/P(G') + 1 + 8ca(
-)2(eg(E
')
+ 1)2. Because G' is obtained from G
after at most p splittings and P is an a-splittable parameter, we have P(G') <
k + ap. Because eg(E') < eg(E) - 1, we obtain bw(G') < 43eg(E)vk + ap + 1 +
8a( )2(eg(E)) 2
Given Condition 2, we apply the induction hypothesis to each of the connected
components of G. Let Gi be such a component. We get that bw(Gi) < 43(eg(E') +
1) /P(G,) + 1 + 8Qa(1)2(eg(E' ) + 1)2. Because Gi is a contraction of some graph G*
obtained from G after at most p splittings and P is an a-splittable parameter, we get
that P(Gi) < P(Gf) < k+cp. Again because eg(E') < eg(E)-1, we have bw(Gi) <
4- eg(E)/k + op + 1+8ca( )2(eg(E))2. Because bw(G') = maxi(bw(Gi)), we obtain
122
bw(G') < 4eg(E)Vk
+ p
1 + 8a(3)2(eg(Z))2.
Because G' is the result of at most p consecutive vertex splittings on G, Lemma 5.9
yields that bw(G) • bw(G') + p. Therefore,
bw(G)
< 43eg()
k + ap + 1 + 8a( ) 2 (eg(E)) 2 + p
< 4eg(E)
k
+ 8a(,)
= 4 eg(E) /
(v+
<
4eg()
4
k+ 1
2 + 4
12)(eg())
8)(eg())
4++
+
/(/k
+1±4a
2 (eg(E)) 2 +
)(+± 1+ 4at)+ 8a( ) 2(eg())2 +4 i + 1,
because a, 36
, >0
= 4eg(E)(Vk
7
1 + 4a) + 8a( )2(eg(E))2 + 4 vk+T
= 4eg(E)vk + 1 + 16a( )2eg(E) + 8a( )2(eg(E))2 + 4vk+
=
4@(eg(E) + I)v/ 4+1+8a(
=
43(eg(E) + 1)v'Tk
1
)2(eg(E)
+ 8a()
2
2(eg(E) 2
I
+ 2eg(E))
+ 2eg(Z) + 1),
because a,/3,6
= 43(eg() + l)vk+T1+ 8a((eg(X) + 1))2.
Theorem 5.10 is a general theorem that applies to any a-splittable bidimensional
parameter.
It is worth mentioning that the "a-splittablity constraint" in Theo-
rem 5.10 can be removed by changing slightly the definition of (contraction) bidimensional parameters for bounded-genus graphs to match with the definition mentioned
in Section 1.4 (see [73] for the details).
For minor-bidimensional parameters, the bound for branchwidth can be further
improved.
Theorem 5.11. Suppose that P is a minor-bidimensionalparameter with density
6 < 1 and normalization factor 3 > 1. Then, for any graph G 2-cell embedded in a
surface E of Euler genus eg(E), bw(G) < 4(eg(E)
+ 1)vP(G) + 1.
Proof. The proof is similar to the proof of Theorem 5.10. The only difference is that,
instead of a fair splitting along the vertices of a minimum-size non-contractible noose,
123
O0
we just remove vertices of the noose from the graph. Because the parameter is minor
bidimensional, the parameter cannot increase by this operation. The rest of the proof
proceeds as before. Let G be a graph 2-cell embedded in a surface E of Euler genus
eg(E), and let k = P(G). We have the following substantially simpler inequality than
the one in Theorem 5.10:
bw(G) < 4eg(E)
5.2.5
+4
+I +p < 4eg(E)
== 4 (eg(E) + 1) k+1
Combinatorial Results and Further Improvements
As a consequence of Theorem 5.11, we establish an upper bound on the treewidth
(or branchwidth) of a bounded-genus graph that excludes some planar graph H as a
minor.
As part of their seminal Graph Minors series, Robertson and Seymour proved the
following:
Theorem 5.12. [144] If G excludes a planar graph H as a minor, then the branchwidth of G is at most bH and the treewidth of G is at most tH, where bH and tH are
constants depending only on H.
The current best estimate of these constants is the exponential upper bound tH <
5
20 2(21V(H)I+4lE(H)f)
[151]. However, it is known that planar graphs can be excluded
"quickly" from planar graphs.
More precisely, the following result says that, for
planar graphs, the constants depend only linearly on the size of H:
Theorem 5.13. [151]If G is planar and excludesa planar graphH as a minor, then
the branchwidthof G is at most 4(21V(H)i + 41E(H)) - 3.
Essentially the same proofs of Theorems 5.10 and 5.11 yield the following generalization of Theorem 4.2 for graphs of bounded genus. In fact, though, we can prove
the following result directly from Theorem 5.11.
124
Theorem 5.14. If G is a graph of Euler genus eg(G) with branchwidth more than
4r(eg(G) + 1), then G has the (r x r)-grid as a minor.
Proof. Consider the parameter
(G) = max{r2 I G has an (r x r)-grid as a minor}.
This parameter never increases when taking minors, and has value r2 on the (r x
r)-grid, so is minor bidimensional with density 1 and normalization factor 1. If
G excludes the (r x r)-grid as a minor, then
(G) < r2 , so (G) < r2
Theorem 5.11, we have that bw(G) < 4(eg(G) + 1)
(G)
-
1. By
< 4(eg(G) + 1)r,
proving the contrapositive of the theorem.
D
Following the proofs of Theorems 5.10 and 5.11 we are also able to quickly exclude any planar graph from bounded-genus graphs. In other words, we generalize
Theorem 5.13 as follows:
Theorem 5.15. If G is a graph of Euler genus eg(G) that excludesa planar graph
H as a minor, then its branchwidthis at most 4(21V(H)I + 4JE(H))(eg(G) + 1).
5.2.6
Algorithmic Consequences
As we already discussed, the combinatorial upper bounds for branchwidth/treewidth
are used for constructing subexponential parameterized algorithms as follows. Let
G be a graph and P be a parameterized problem we need to solve on G. First one
constructs a branch/tree decomposition of G that is optimal or "almost" optimal.
A (, y, A)-approximation scheme for branchwidth/treewidth
consists of, for every
w, an 0(27wnA)-time algorithm that, given a graph G, either reports that G has
branchwidth/treewidth
at least w or produces a branch/tree decomposition of G with
width at most Ow. For example, the current best schemes are a (3 + 2/3, 3.698, 3 + e)approximation scheme for treewidth [11] and a (3, lg 27, 2)-approximation scheme for
branchwidth
[148].
If the branchwidth/treewidth
of a graph is "large", then combinatorial upper
bounds come into play and we conclude that P has no solution on G. Otherwise we
run a dynamic program on graphs of bounded branchwidth/treewidth
P(G).
125
and compute
Thus we conclude with the main algorithmic result of this section:
Theorem 5.16. Let P be a bidimensionalparameterwith density 6 and normalization
factor f. SupposeP is either minor bidimensional,in which casewe set PI= O,or P is
contraction bidimensionaland a-splittable,in which case we set iL= 2. Suppose that
there is an algorithmfor the associatedparameterizedproblemthat runs in 0(2awnb)
time given a tree/branch decomposition of the graph G with width w. Suppose also
that we have a (, y, A)-approximation scheme for treewidth/branchwidth. Set T = 1 in
the case of branchwidth and
T
= 1.5 in the case of treewidth. Then the parameterized
problem asking whether P(G) < k can be solved in
o( 2maxa0,}y)4 (g(G)+1)(\JkT+((G)+))nmax{b,} ) time.
The existence of an 0(2"anb)-time algorithm for treewidth/branchwidth
w holds
for many examples of bidimensional parameters with small values of a and b; see
[2, 6, 50, 74, 93, 123, 94]. Observe that the correctness
of our algorithms
is simply
based on Theorems 5.10 and 5.11, despite their nonalgorithmic natures, and (, y, A)approximation scheme for branch/tree decomposition. We note that the time bounds
we provide do not contain any hidden constants, and the constants are reasonably low
for a broad collection of problems covering all the problems for which 2°(V)n°()-time
algorithms already exist.
5.3 H-Minor-Free Graphs
In this section we demonstrate how the results on graphs of bounded genus can be
generalized on graphs with excluded minors. More precisely we show that, given
the tree decompositions guaranteed by Theorems 1.3 and 1.4, we can obtain efficient
algorithms for problems on H-minor-free graphs. Although our main development
is in terms of dominating set, our approach can be viewed as a guideline for solving
other problems on H-minor-free graphs. Some further results in this direction are
described in Section 5.4.
126
5.3.1
Almost-Embeddable Graphs and r-Dominating Set
In order to treat each term separately in the clique-sum decomposition of an Hminor-free graph, we need to solve a more general problem than dominating set. This
r-dominating set problem, which also arises in facility location, is also contraction-
bidimensional. This property enables us to obtain a parameter-treewidth bound for
this problem as well.
Definition
5.17. Let G be a graph. A subset D C V(G) of vertices r-dominates
another subset S C V(G) of vertices if each vertex in S is at distance at most r from
a vertex in D. We say that D is an r-dominating set if it r-dominates V(G).
We need the following result proved in [64].
Lemma 5.18. ([64], see also Chapter 4) Let p, k, r > 1 be integers and G be a planar
graph having an r-dominating set of size k and containing a (p x p)-grid as a minor.
Then k > (e)2
In other words, Lemma 5.18 says that, for any fixed r, r-dominating set is a bidimensional parameter. It is also easy to see that it is 1-splittable. Thus Theorem 5.10
yields the following lemma.
Lemma 5.19. For any constant r, if a graph G of genus g has an r-dominating set
of size at most k, then the treewidth of G is O(gxV + g 2).
Now we extend this result to apex-free h-almost-embeddable graphs. Before expressing this result, we need the following slight modification of [103, Lemma 2].
Lemma 5.20. Let G = Go UG1 U -UGh be an apex-free h-almost-embeddable graph.
For 1 < i < h, let (u)uEu
be the path decomposition of vortex Gi of width at most h.
Suppose that, for each 1 < i < h, the vertices Ui = ful, u
..
um form a path in
Go. Then tw(G) < (h 2 + 1)(tw(Go) + 1) - 1.
Proof. Let B be a bag of a tree decomposition of Go of minimum width tw(Go). For
each index 1 < i < h, and for each vertex u E B n Ui, we add to B the corresponding
127
bag Bu of the path decomposition of Gi. The size of each Bu is at most h, and the
original size of B is also at most tw(Go) + 1. Thus such additions increase the size
of B by at most h 2 (tw(Go) + 1). Performing these additions for all bags B of a tree
decomposition increases the maximum bag size from tw(Go)+ 1 to (h2 + 1)(tw(Go) +
1). It can be easily seen that the resulting set of bags B form a tree decomposition
of G, because each Ui forms a path in Go.
O
Lemma 5.21. Let r be a constant and let G = Go U G1 U ... U Gh be an apexfree h-almost-embeddable graph on a surface E of genus g. Let k be the size of a
set D C V(G) that r-dominates V(Go). Then tw(G) = O(h2 (gv/k'T+h+ g2 )). In
particular,for fixed g and h, tw(G) = O(
Proof. For each 1
i
).
h, let (Bu)Ueui be the path decomposition of vortex Gi,
where Ui = {u , u?,... , ui.
Let Go be the graph obtained from Go by adding new
vertices C = ({c, C2 , ·
Ch} and edges (Ci,u j ) and (u, u+l) (where j + 1 is treated
modulo mi) for all 1
i < h and 1
Go is also embeddable in
j < mi. Because Go is embeddable in A,
. G has an r-dominating set of size at most k + h,
namely, (D n V(Go)) U C. By Lemma 5.19, tw(G') = O(gvfFkVI + g 2 ). Also, in
Go, the vertices Ui, 1
i
h, form a path.
G' = G U G1 U ... U Gh is O(h 2 (g v~kT
By Lemma 5.20, the treewidth of
+ g 2 )). Finally, because G is a subgraph of
°
G', tw(G) < tw(G').
5.3.2
H-Minor-Free Graphs and Dominating Set
Now that we have an understanding of r-dominating set in apex-free almost-embeddable
graphs, we return to the original problem of dominating set in the more general setting of H-minor-free graphs. For this section we use the notation G* for the entire
H-minor-free graph so that the primary object of interest, an almost-embeddable
piece of G*, can be referred to as G. The main result of this section is the following
algorithmic result.
Theorem 5.22. One can test whether an H-minor-free graphG* has a dominating
set of size at most k in time 20(V~)n0(1), where the constants in the exponents depends
128
on H.
Before mentioning the proof of the above theorem, we need some definitions and
lemmas.
Definition 5.23. Consider a clique-sum decompositionof an H-minor-free graph
G* according to Theorem 1.3, organized into a tree structure (T, X) as described in
Section 1.3.3. Let G be one term in the clique-sum decomposition of G* that is an
h-almost embeddableon a surface of genus g, with apex set X. If we remove from
T the node of T correspondingto term G, we obtain a forest T' of p subtrees; let
G 1, G2 , . . , Gp denote the clique-sums of the terms corresponding to the nodes in each
connectedcomponentof T'. We say that G is clique-summedwith eachGi, I
with join set Wi = V(G) n V(Gi).
i < p,
Because the clique-sums are at most h-sums,
[Wil < h. A clique Wi is called fully dominated by a subset S C V(G) of vertices in
G if V(Gi) - X C NG.(S); otherwise, clique Wi is called partially dominated by S.
A vertex v of G is fully dominated by a set S if NG*[(G)-X] [V]C NG*(S).
Note that the only edges that can appear in G but not in G* are the edges among
vertices of Wi, 1 < i < p.
Theorem 5.24. Let G be an h-almost embeddableon a surface of genusg in a cliquesum decomposition of a graph G*. Suppose G is clique-summed with graphs G1, . ,
via join sets W1 ,..., Wp, where IWi < h, 1
Gp
i < p. SupposeG* has a dominatingset
of size at most k. Then there is a subset S C V(G) of size at most h such that, if we
form the graphG by removing allfully dominatedvertices that are not includedin any
partially dominated clique Wi from G, then tw(G) = O(h2gviT
+ 92 ) = 0(
).
Proof. Suppose X is the set of apices in G, so that G - X is an apex-free h-almost
embeddable graph. Let D be a dominating set of size k of G* and let S = X n D. We
claim that S is our desired set. The rest of the proof is as follows: we construct a set
D of size at most k for G - X which 2-dominates every vertex v of G - X which is
not included in any vortex. Then since G - X is an apex-free h-almost-embeddable
on a surface of genus g with a 2-dominating-type set of size at most k desired by
129
Lemma 5.21, it has treewidth at most O(h 2 gvfk
+ g2). Then we can add vertices
of X to all bags and still have a tree decomposition of width O(h 2gx/ T-+h + g 2),
as desired. We construct D from D as follows. First, we set D = D n V(G). For
each 1
i < p, if D n (V(Gi)-
Wi) - 0 and Wi g X, we add an arbitrary vertex
w E Wi - X to D. Here we say a vertex v of D is mapped to a vertex w of D if v = w
or if v E D n (V(Gi) - Wi) and vertex w E Wi - X is the one that we have added
to D. One can easily observe that since each new vertex in D is in fact accounted
by a unique vertex in D,
bi1<
k. It only remains to show that D is a 2-dominating
set for G - X. If a vertex v E V(G) - X is not fully-dominated, then there exists a
vertex w E NG[V]which is not dominated by S and thus not dominated by X (since
S = D n X). It means v is 2-dominated by a vertex u of G - X which dominates w
(we note that u can be originally a vertex u' in (V(Gi) - Wi) n D which is mapped
to u in D). Also, we note that for each clique Wi in which there is a mapped vertex
of D, this vertex dominates all vertices of Wi - X in G - X and thus we keep the
whole clique Wi - X in G. It only remains to show that every vertex of a partially
dominated clique Wi is 2-dominated by a vertex of G - X. We consider two cases: if
WinS = 0, since V(Gi)-Wi
0, there must exists a (mapped) vertex of D in Wi-X
and we are done. Now assume Wi n s # 0. If Wi c X then Wi n (V(G) - X) = 0
and we are done (since there is no clique in G - X at all.) Otherwise, there exists
a vertex Wi - X. If (V(Gi) - Wi) C NG(S) 7 3 0, then V(Gi) n D
0. Thus there
exists a mapped vertex w E Wi - X and we have 1-dominated vertices of Wi - X. As
mentioned before if D n (Wi - X)
0, vertices Wi - X are 1-dominated and we are
done. The only remaining case is the case in which there exists a vertex w E Wi - X
which is dominated by a vertex x E V(G) and by assumption w V NG (S) (we note
that in this case, there is no dominating vertex in V(Gi) - Wi for any i for which
w E Wi.) It means vertex x is not fully dominated and thus it remains in G. In
addition, vertex x 2-dominates all vertices of Wi - X, since Wi is a clique in G and
thus all vertices of Wi - X are 2-dominated. This completes the proof of the theorem.
Now, we are ready to prove Theorem 5.22.
130
Proof of Theorem 5.22. First, we use the n°()-time
algorithm of Theorem 1.4 to
obtain the clique-sum decomposition of graph G*. As mentioned before, this cliquesum decomposition can be considered as a generalized tree decomposition of G*.
More precisely, we consider the clique-sum decomposition as a rooted tree. We
try to find a dominating set of size at most k in this graph using a two-level dynamic
program. Suppose a graph G is an h-almost-embeddable graph on a surface of genus
g in a clique-sum decomposition of a graph G*. Assume G is clique-summed with
graphs Go0 G1 , .... , Gp via join sets Wo, W, . .., Wp, where Wil < h, O < i < p. Also
assume that Go is considered as the parent of G and G 1 ,..., Gp are considered as
children of G.
Colorings.
The subproblems in our first-level dynamic program are defined by a
coloring of the vertices in Wi. Each vertex will be assigned one of 3 colors, labelled 0,
T1, and l 1. The meaning of the coloring of a vertex v is as follows. Color 0 represents
that vertex v belongs to the chosen dominating set. Colors
1 and T 1 represent
that the vertex v is not in the chosen dominating set. Such a vertex v must have
a neighbor w in the dominating set (i.e., colored 0); we say that vertex w resolves
vertex v. Color
1 for vertex v represents that the dominating vertex w is in the
subtree of the clique-sum decomposition rooted at the current graph G, whereas T1
represents that the dominating vertex w is elsewhere in the clique-sum decomposition.
Intuitively, the vertices colored
1 have already been resolved, whereas the vertices
colored T1 still need to be assigned to a dominating vertex.
Locally valid colorings.
A coloring of the vertices of Wi is called locally valid with
respect to sets S1 , S 2 C V(G) if the following properties hold:
* for any two adjacent vertices v and w in Wi, if v is colored 0, w is colored
and
* if v E S1 n Wi, then v is colored 0; and
* if v E
S2
n Wi, then v is not colored 0.
131
11;
Our colorings are similar to that of previous work (e.g., [2]), but we use them
in a new dynamic-programming framework that acts over clique-sum decompositions
instead of tree decompositions.
Dynamic program subproblems.
Our first-level dynamic program has one sub-
problem for each graph G in the clique-sum decomposition and for each coloring
c of the vertices in W0. Because each join set has at most h vertices, the number of subproblems is O(n
3 h).
We define D(G, c) to be the size of the minimum
"semi"-dominating set of the vertices in subtree rooted at G subject to the following
restrictions:
1. Vertices colored
1 are adjacent to at least one vertex in the dominating set.
(Vertices colored 1 are dominated "for free".)
2. Vertices colored 0 are precisely the vertices in the dominating set.
3. Vertices in Wo are colored according to c.
If we solve every such subproblem, then in particular, we solve the subproblems
involving the root node of the clique-sum decomposition and in which every vertex
is colored 0 or 1. The final dominating set of size k is given by the best solution to
these subproblems.
Induction step.
Suppose for each coloring c of Wi, 1 < i < p, we know D(Gi, c). If
the graph G is of size at most h, then we can try all colorings in O( 3 h h2) = 0(1) time
(where the factor of h2 is for checking validity). Thus, we focus on almost-embeddable
graphs G. First, we guess a subset X of size at most h. Then for each subset S of X,
we put the vertices of S in the dominating set and forbid vertices of X - S from being
in the dominating set. Now we remove from G all fully dominated vertices of G - X
that are not included in any partially dominated clique Wi. Call the resulting graph
G. By Theorem 5.24, tw(G) = O(Vx/). We can obtain such a tree decomposition of
width 3+2/3 times optimum, in 2°(v')n3+e time by a result of Amir [11]. All vertices
absent from this tree decomposition are fully dominated and thus, in any minimum
132
dominating set that includes S, they will not appear except the following case. It
is possible that up to IX - SI = O(h) vertices, which are either fully dominated or
belong to V(Gi) - Wi where Wi is fully dominated, appear in the dominating set to
dominate vertices of X - S. Call the set of such vertices S'. We can guess this set
S' by choosing at most h vertices among the discarded vertices that have at least
one neighbor in X - S, and then add S' to the dominating set. On the other hand,
for any partially dominated clique Wi, we know that all of its vertices are present
in the tree decomposition; because they form a clique, there is a bag ai in any tree
decomposition that contains all vertices of Wi. We find ai in our tree decomposition
and map Wi and Gi to this bag. We also assume W0ois contained in all bags, because
its size is at most h. Now, for each coloring c of W0o,we run the dynamic program
of Alber et al. [2] on the tree decomposition, with the restriction that the colorings
of the bags are locally valid with respect to S1 := S U S' and S2 := X - S, and are
consistent with the coloring c of W0 . For each bag ai to which we mapped Gi, we
add to the cost of the bag the value D(Gi, c') for the current coloring c' of Wi. Using
this dynamic program, we can obtain D(G, c) for each coloring c of W0 .
Running time.
The running time for each coloring c of W0 and each choice of S
is 2°(V)n according to [2]. We have
3h
choices for c, O(nh+ l ) choices for X, 0( 2 h)
choices for S, and O(nh+l ) choices for S'. Thus the running time for this inductive
step is
6hn2h+2 2
0("V) . There are O(n) graphs in the clique-sum decomposition of G.
Therefore, the total running time of the algorithm is 0( 6hn2h+3
2
°(V )) + n° (1) (the
latter term for creating the clique-sum decomposition), which is 2°(V)n°(1) as desired.
5.4
Concluding Remarks
We have shown how to obtain subexponential fixed-parameter algorithms for the
broad class of bidimensional problems on bounded-genus graphs, and for dominating
set on general H-minor-free graphs for any fixed H. Our approach can also be used
133
to obtain subexponential algorithms for other problems on H-minor-free graphs. We
now demonstrate some examples of such problems.
The first example is vertex cover, where we use the following reduction.
For
a graph G, let G' be the graph obtained from G by adding a path of length two
between any pair of adjacent vertices. The following lemma is obvious.
Lemma 5.25. For any Kh-minor-free graphG, h > 4, and integerk > 1,
* G' is Kh-minor-free, and
* G has a vertex cover of size < k if and only if G' has a dominating set of size
< k.
Combining Lemma 5.25 with Theorem 5.22, we conclude that parameterized vertex cover can be solved in subexponential time on graphs with an excluded minor.
Another example is the set-cover problem. Given a collection C = {(C, C2, · · , Cm}
of subsets of a finite set S = {(s, s2,..., sn), a set cover is a subcollection C' C C
such that UcIEc' Ci = S. The minimum set cover problem is to find a cover of minimum size. For an instance (C, S) of minimum set cover, its graph Gs is a bipartite
graph with bipartition (C, S). Vertices si and Cj are adjacent in Gs if and only if
si E Cj. Theorem 5.22 can be used to prove that minimum set cover can be solved in
subexponential time when Gs is H-minor free for some fixed graph H. Specifically,
for a given graph Gs, we construct an auxiliary graph As by adding new vertices
v, u, w and making v adjacent to {u, w, C1 , C2 ,..
.,
Cm}. Then
* (C, S) has a set cover of size < k if and only if As has a dominating set of size
< k + 1, and
* if Gs is Kh-minor-free, then As is Kh+l-minor-free.
It is reasonable to believe that Theorem 5.22 generalizes to obtain a subexponential fixed-parameter algorithm for the (k, r)-center problem on H-minor-free graphs
Recall that the (k, r)-center problem is a generalization of the dominating-set problem in which the goal is to determine whether an input graph G has at most k
134
vertices (called centers) such that every vertex of G is within distance at most r
from some center. In Chapter 4, we considered this problem for planar graphs and
map graphs, and presented a generalization of dynamic programming mentioned in
the proof of Theorem 5.22 to solve the (k, r)-center problem for graphs of bounded
treewidth/branchwidth.
This dynamic program and Theorem 5.24 can be generalized
to establish the desired result for H-minor-free graphs. A consequence is that we can
solve the dominating-set problem in constant powers of H-minor-free graphs, which
is the most general class of graphs so far for which one can obtain the exponential
speedup.
It is an open and tempting question whether our technique can be generalized
to solve in subexponential time on H-minor-free graphs every problem that can be
solved in subexponential time on bounded-genus graphs. Recent positive progress
on this question has been made [71] (see Chapter 4). Based on our results, one
can obtain subexponential algorithms for any minor-bidimensional problem on Hminor-free graphs, and for any contraction-bidimensional problem on apex-minorfree graphs. Note that these results, while general, cannot be applied directly to
dominating set on H-minor-free graphs. In particular, it remains open to extend the
algorithmic approaches of Section 5.3 for H-minor-free graphs to all bidimensional
parameters.
We also suspect that there is a strong connection between bidimensional parameters and the existence of linear-size kernels for the corresponding parameterized
problems in bounded-genus graphs. Such a linear kernel has recently been obtained
for dominating
set [94].
135
136
Chapter 6
Diameter and Treewidth in
Minor-Closed Graph Families
Eppstein [87] introduced the diameter-treewidth property for a class of graphs, which
requires that the treewidth of a graph in the class is upper bounded by a function
of its diameter. This notion has been used extensively in a slightly modified form
called the bounded-local-treewidthproperty, which requires that the treewidth of any
connected subgraph of a graph in the class is upper bounded by a function of its
diameter. For minor-closed graph families, which is the focus of most work in this
context, these properties are identical.
The reason for introducing graphs of bounded local treewidth is that they have
many similar properties to both planar graphs and graphs of bounded treewidth,
two classes of graphs on which many problems are substantially easier. In particular, Baker's approach for polynomial-time approximation schemes (PTASs) on planar graphs [23] applies to this setting. As a result, PTASs are known for hereditary
maximization problems such as maximum independent set, maximum triangle matching, maximum H-matching, maximum tile salvage, minimum vertex cover, minimum
dominating set, minimum edge-dominating set, and subgraph isomorphism for a fixed
pattern [72, 87, 110]. Graphs of bounded local treewidth also admit several efficient
fixed-parameter algorithms. In particular, Frick and Grohe [96] give a general framework for deciding any property expressible in first-order logic in graphs of bounded
137
local treewidth.
The foundation of these results is the following characterization by Eppstein [87]
of minor-closed families with the diameter-treewidth property. Recall from Chapter 1
that an apex graph is a graph in which the removal of some vertex leaves a planar
graph.
Theorem 6.1. Let . be a minor-closed family of graphs. Then . has the diametertreewidth property if and only if Y does not contain all apex graphs, i.e.,
excludes
some apex graph.
In this Chapter, we reprove this theorem with a much simpler proof. Similar
to Eppstein's proof, we use the following theorems from Graph Minor Theory. The
m x m grid is the planar graph with m2 vertices arranged on a square grid and with
edges connecting horizontally and vertically adjacent vertices.
Theorem 6.2 ([80]). For integers r and m, let G be a graph of treewidth at least
m 4 r 2(m+2).
Then G contains either the complete graph Kr or the m x m grid as a
minor.
It is worth mentioning that we improve the bound in the theorem above in Chapter 8.
Theorem 6.3 ([151]). Every planar graph H can be obtained as a minor of the r x r
grid H, wherer = 141V(H)I- 24.
As we show in Chapter 7, the results of this chapter makes the foundation of the
bidimensionality theory in apex-minor-free graphs.
Before presenting the proof of Theorem 6.1, first we present a property of apexminor-free graphs. The vertices (i,j) of the m x m grid with i E {1, m} or j E
{1, m} are called boundary vertices, and the rest of the vertices in the grid are called
nonboundary vertices.
Lemma 6.4. Let G be an H-minor-free graph for an apex graph H, let k = 141V(H)l22, and let m > 2k be the largest integer such that tw(G) >
138
m41V(H)12 (m+2).
Then
G can be contracted into an augmented grid R, i.e., a (m - 2k) x (m - 2k) grid
augmented with additional edges (and no additional vertices) such that each vertex
v E V(R) is adjacent to less than (k + 1)6 nonboundary vertices of the grid.
Proof By Theorem 6.2, G contains an m x m grid M as a minor. Thus there exists
a sequence of edge contractions and edge/vertex deletions reducing G to M.
We
apply to G the edge contractions from this sequence; we ignore the edge deletions;
and instead of deletion of a vertex v, we only contract v into one of its neighbors.
Call the new graph G', which has the m x m grid M as a subgraph and in addition
V(G') = V(M).
We claim that each vertex v E V(G') is adjacent to at most k4 vertices in the
central (m - 2k) x (m - 2k) subgrid M' of M. In other words, let N be the set of
neighbors of any vertex v E V(G') that are in M'. We claim that INI < k4 . Suppose
for contradiction
that
INI > k4.
Let n. denote the number of distinct x coordinates of the vertices in N, and let ny
denote the number of distinct y coordinates of the vertices in N. Thus, INI < n, n .
Assume by symmetry that ny > n=, and therefore ny >
(NI > k 2.
We define the subset N' of N by removing all but one (arbitrary chosen) vertex
that share a common y coordinate, for each y coordinate. Thus, all y coordinates of
the vertices in N' are distinct, and IN'[ = ny > k2 . We discard all but k2 (arbitrarily
chosen) vertices in N' to form a slightly smaller set N". We divide these k 2 vertices
into k groups each of exactly k consecutive vertices according to the order of their
y coordinates. Now we construct the minor k x k grid K as shown in Figure 6-1.
Because each y coordinate is unique, we can draw long horizontal segments through
every point. The k columns on the left-hand and right-hand sides of M allow us to
connect these horizontal segments together into k vertex-disjoint paths, each passing
through exactly k vertices of N". These paths can be connected by vertical segments
within each group. This arrangement of paths has the desired k x k grid K as a
minor, where the vertices of the grid correspond to the vertices in N".
Now, if v has been used in the contraction of a vertex v' in K, we proceed as
shown in Figure 6-2. First we "lift" v' from the grid-not
139
removing it from the graph
k
k
Figure 6-1: Construction of the minor k x k grid K.
per se, but marking it as "outside the grid." Then we contract the remainder of v"s
column and the two adjacent columns (if they exist) into a single column. Similarly we
contract the remainder of v"s row with the adjacent rows. Thus we obtain as a minor
of K a (k - 2) x (k - 2) grid K' such that vertex v' is outside this grid and adjacent
to all vertices of the grid. Now, by Theorem 6.3, because k- 2 > 141V(H)I - 24, we
can consider v' as the apex of H and obtain the planar part of H as a minor of K'.
Hence the original graph G is not H-minor-free, a contradiction. This concludes the
proof of the claim that NI < k4.
-
(a)
-
-
-`
-
(b)
(c)
(d)
Figure 6-2: In the k x k grid K, we (a) lift the vertex v', (b) contract the adjacent
columns, and (c) contract the adjacent rows, to form a (k - 2) x (k - 2) grid K'.
Vertex v' is adjacent to all vertices in the grid, though the figure just shows four
neighbors for visibility.
Finally, form a new graph R by taking graph G' and contracting all 2k boundary
rows and 2k boundary columns into two boundary rows and two boundary columns
(one on each side). The number of neighbors of each vertex of R that are not on the
boundary is at most (k + 1)2 k4 . The factor (k + 1)2 is for the boundary vertices each
of which is obtained by contraction of at most (k + 1)2 vertices.
Now we are ready to prove Theorem 6.1.
140
0
of Theorem 6.1. One direction is easy. The apex graphs Ai, i = 1, 2,...,
obtained
from the i x i grid by connecting a new vertex v to all vertices of the grid have diameter
two and treewidth i+ 1, because the treewidth of the i x i grid is i (see e.g. [79]). Thus
a minor-closedfamily of graphs with the diameter-treewidth property cannot contain
all apex graphs. Next consider the other direction. Let G be a graph from a minorclosed family F of graphs excluding an apex graph H. We show that the treewidth of
G is bounded above by a function of IV(H)I and its diameter d. Let m be the largest
integer such that tw(G)
(m-2k) x (m-2k)
> m41V(H)1 2 (m+2),
and let k = 141V(H)I - 22. Let R be the
augmented grid obtained from G by contraction, using Lemma 6.4.
Because diameter does not increase by contraction, the diameter of R is at most d. In
addition, one can easily observe that the number of vertices of distance at most i from
any vertex in R is at most 4r+4r(k+1) 6 +4r(k+1) 1 2 +. ..+4r(k+1) 6i < 4r(k+1)6 (i+l),
where r = m - 2k. Because the diameter is at most d, we have 4r(k + 1)6(d+
l
) > r2
i.e., m < 2k + 4(k + 1)6(d+1). Thus the treewidth of G is at most (4(k + 1)6(d+1) +
2k + 1)41V(H)12(4(k+1)6(d+l)+2k+3) = O(IV(H)
6(d+l))O(IV(H)16d+8 ) =
as desired.
2 0(dlV(H)1
6
d+s1gV(H)I),
O
141
142
Chapter 7
Bidimensional Parameters and
Local Treewidth
As mentioned in previous chapters, the majority of results for designing FPT algorithms to solve problems such as k-vertex cover or k-dominating set in a sparse graph
class F are based on the following lemma: every graph G in F where the value of the
graph parameter is at most k has treewidth bounded by t(k), where t is a strictly increasing function depending only on F. With some work (sometimes very technical),
a tree decomposition of width t(k) is constructed and standard dynamic-programming
techniques on graphs of bounded treewidth are implemented. Of course this method
can not be applied for any graph class F. For instance, the n-vertex complete graph
K, has a dominating set of size one and treewidth equal to n - 1. So the emerging
question is:
For which (larger) graph classes and for which graph parameters can the
"bounding treewidth method" be applied?
In this chapter we give a complete characterization of minor-closed graph families
for which the aforementioned "bounding treewidth method" can be applied for a
wide family of graph parameters.
More precisly, we show that for a large family
of contraction-bidimensional graph parameters, a minor-closed graph family F has
the parameter-treewidth property if F has bounded local treewidth. Moreover, we
143
show that the inverse is also correct if some simple condition is satisfied by P. In
addition we show that, for a slightly smaller family of minor-bidimensional graph
parameters, every minor-closed graph family F excluding some fixed graph has the
parameter-treewidth property.
The proof of the main result uses the characterization of Eppstein for minorclosed families of bounded local treewidth [87] (or its simplification in Chapter 6)
and Diestel et al.'s modification of the Robertson & Seymour excluded-grid-minor
theorem [80]. In addition, the proof is constructive and can be used for constructing
fixed-parameter algorithms to decide bidimensional graph parameters on minor-closed
families of bounded local treewidth.
These algorithms parallel the general fixed-
parameter algorithm of Frick and Grohe [96] for properties definable in first-order
logic in graph families of bounded local treewidth; our results apply e.g. to minorbidimensional parameters definable in monadic second-order logic in nontrivial minorclosed graph families. See Section 7.4 for details.
This chapter is organized as follows. Section 7.1 contains the formal definitions of
the concepts used in the chapter. Section 7.2 presents two combinatorial results which
support the main result of the chapter, proved in Section 7.3. Finally, in Section 7.4
we present the algorithmic consequences of our results and we conclude with some
open problems.
7.1 Definitions and Preliminary Results
We first need the following facts about treewidth. The first fact is trivial.
* For any complete graph Kn on n vertices, tw(Kn) = n - 1.
The second fact is well known but its proof is not trivial. (See e.g., [79].)
* The treewidth of the m x m grid is m.
The next fact we need is a theorem on excluded grid minors due to Diestel et
al. [80]. (See also the textbook
[79].)
144
Theorem 7.1 ([80]). Let r, m be integers, and let G be a graph of treewidthat least
m4r2 (m +2).
Then G contains either Kr or the m x m grid as a minor.
Recall that a graph parameter P is a function mapping graphs to nonnegative
integers. The parameterized problem associated with P asks, for a fixed k, whether
P(G) < k for a given graph G. Given a graph parameter P, we say that a graph
family F has the parameter-treewidthproperty for P if there is a strictly increasing
function t such that every graph G E F has treewidth at most t(P(G)).
Definition 7.2. Let g: N - N be a strictly increasing function. We say that a graph
parameterP is g-minor-bidimensional if
* Contractingan edge, deleting an edge, or deletinga vertex in a graphG cannot
increaseP(G).
* For the r x r grid R, P(R) > g(r).
Similarly, a graphparameter P is g-contraction-bidimensionalif
* Contractingan edgein a graphG cannot increaseP(G).
* For any r x r augmentedgrid R of constant span, P(R) > g(r).
In the above definition, an r x r augmentedgrid of span s is an r x r grid with
some extra edges such that each vertex is attached to at most s non-boundary vertices
of the grid (see an example in Figure 7-1). Intuitively, bidimensional parameters are
required to be "large" in two-dimensional grids.
We note that a g-minor-bidimensional parameter is also a g-contraction-bidimensional
parameter.
One can easily observe that many graph parameters such as minimum
sizes of dominating set, q-dominating set (distance q-dominating set for a fixed q),
vertex cover, feedback vertex set, and edge-dominating set (see exact definitions of
the corresponding graph parameters in [74]) are O(r 2 )-minor- or O(r 2 )-contractionbidimensional parameters.
1Closelyrelated notions of bidimensional parameters are introduced by the authors in [63].
145
Figure 7-1: An augmented 12 x 12 grid with span 8.
Here, we present a theorem for minor-bidimensional parameters on general minorclosed classes of graphs excluding some fixed graphs, which plays an important role
in the main result of this chapter.
Theorem 7.3. If a g-minor-bidimensionalparameter P on an H-minor-free graph
G has value at most k, then tw(G) <
41
2
g2 v(H) (9-l(k)+2)log(
(k)) = 2 0(g- (k) log( -(k)))
Proof. Notice that KIV(H)lcontains as a minor any graph on IV(H)I vertices. Therefore we may assume that G is KIV(H)I-minor-free.If the claimed upper bound for the
treewidth of G is not correct, then Theorem 7.1 implies that G contains a m x m grid R
as a minor for m > g-l(k). Because P is g-minor-bidimensional, P(R) > g(m). The
bidimensionality of P along with the fact that R is a minor of G yield P(G) > g(m).
Therefore, k > g(m), a contradiction.
oE
Theorem 7.3 can be applied for minor-bidimensional parameters such as vertex
cover or feedback vertex set.
7.2
Combinatorial Lemmas
In this section we prove two combinatorial lemmas regarding grids and graphs of
bounded local treewidth.
The proof of the following lemma uses the ideas in the
proof of Lemma 6.1.
146
..j :::
.
.... .. ..
.... . ...
"ii
.. .. .. ... .... ...... 1
........ .. .............
.........
...... ..
. . ............
-.i ;i
.....
~~.;......~~~_..:--
---------...
..
.....
....I i
Figure 7-2: Left: The grid H, the points in S", and their grouping. Here = 6.
Right: Construction of the minor x grid R passing through the points in S".
Lemma 7.4. Suppose we have an m x m grid H and a subset S of vertices in the
central (m - 2e) x (m - 2£) subgrid H', where s = ISI and £ = L[-J. Then H has as
a minor the x
grid R such that each vertex in R is a contraction of at least one
vertex in S and other vertices in H.
Proof. Let sx denote the number of distinct x coordinates of the vertices in S, and let
sy denote the number of distinct y coordinates of the vertices in S. Thus, s < sx . sy.
Assume by symmetry that sy > s, and therefore sy > V/.
We define the subset S' of S by removing all but one point that share a common
y coordinate, for each y coordinate. Thus, all y coordinates of the vertices in S'
are distinct, and IS' = sy. We discard all but £2 vertices in S' to form a slightly
smaller set S", because S' = s >
S
> (L,fJ) 2
=
2.
We divide these £2 vertices
into e groups each of exactly e consecutive vertices according to the order of their y
coordinates. Now we have the situation shown on the left of Figure 7-2.
We construct the minor grid R as shown on the right of Figure 7-2. Because
each y coordinate is unique, we can draw long horizontal segments through every
point. The
columns on the left-hand and right-hand sides of H allow us to connect
these horizontal segments together into
exactly
vertex-disjoint paths, each passing through
vertices of S". These paths can be connected by vertical segments within
each group. By contracting each horizontal segment into a single vertex, and some
further contraction, we can obtain the desired
147
x
grid R as a minor. Each vertex
of this grid R is a contraction of at least one vertex in S" (and hence in S) and other
vertices in H.
o
Lemma 7.5. Let G E L(f) be a graph containing the m x m grid H as a subgraph,
m > 2, wheref = f(2) + 1. Then the central(m - 2e) x (m - 2e) subgridH' has the
property that every vertex v E V(G) is adjacent to less than £4 vertices in H'.
Proof. Suppose for contradiction that there is a vertex v E V(G) such that S =
NG(v) n V(H) has size s = ISI >_ 4. By Lemma 7.4, H' has as a minor a e x grid
R such that each vertex in R is a contraction of at least one vertex in S and other
vertices in H'. If we perform these contractions and deletions in G, then v is adjacent
to all vertices in R. Define R + v to be the grid R plus the vertex v (if v is not already
in R) and the star of connections between v and all vertices in R. Then R + v is a
minor of G, but has diameter 2 and treewidth
7.3
> f(2) + 1, a contradiction.
O
Main Theorem
Now we are ready to present the main result of this chapter.
Theorem
7.6. Let P be a contraction-bidimensional parameter.
A minor-closed
graph class F has the parameter-treewidthproperty for P if F is of bounded local
treewidth. In particular,for any g-contraction-bidimensionalparameter P, function
f: N tw(G)
N and any graph G E
2 0(9g(k)logg
(k)).
(f) on which P has value at most k, we have
(The constant in the 0 notation depends on f(1) and
f(2).)
Proof. Let r = f(1)+ 1 and f = f(2)+1. Let G E £(f) be a graph on which the graph
parameter P has value k. Let m be the largest integer such that tw(G) >
m4r 2 (m+2).
Without loss of generality, we assume G is connected, and m > 2 (otherwise, tw(G)
is a constant because both r and e are constants.) Then G has no complete graph Kr
as a minor. By Theorem 7.1, G contains an m x m grid H as a minor. Thus there
exists a sequence of edge contractions and edge/vertex deletions reducing G to H.
We apply to G the edge contractions from this sequence, we ignore the edge deletions,
148
and instead of deletion of a vertex v, we only contract v into one of its neighbors.
Call the new graph G', which has the m x m grid H as a subgraph and in addition
V(G') = V(H). Because graph parameter P is contraction-bidimensional, its value
on G' will not increase. By Lemma 7.5, we know that the central (m - 2f) x (m - 2f)
subgrid H' of H has the property that every vertex v E V(G') is adjacent to less than
£4
vertices in H'.
Now, suppose in graph G', we further contract all 2f boundary rows and 2f bound-
ary columns into two boundary rows and two boundary columns (one on each side)
and call the new graph G". Note that here G" and H' have the same set of vertices.
The degree of each vertex of G" to the vertices that are not on the boundary is at
most (f + 1) 2 f 4 , which is a constant because f is a constant. Here the factor (+ 1)2 is
for the boundary vertices each of which is obtained by contraction of at most (f + 1)2
vertices. Again because graph parameter P is contraction-bidimensional, its value on
G" does not increase and thus it is at most p. On the other hand, because the graph
parameter is g-contraction-bidimensional, its value on graph G" is at least g(m - 2f).
Thus g-1 (k) > m - 2, so m = O(g-'(k)).
By Theorem 7.3, the treewidth of the
original graph G is at most
as desired.
2
0(9-l(k)logg9-(k))
The apex graphs Ai, i = 1, 2, 3,...,
[
are obtained from the i x i grid by adding a
vertex v adjacent to all vertices of the grid. It is interesting to see that, for a wide
range of graph parameters, the inverse of Theorem 7.6 also holds.
Lemma 7.7. Let P be any contraction-bidimensionalparameter where P(Ai) = 0(1)
for any i > 1. A minor-closedgraphclass F has the parameter-treewidthpropertyfor
P only if F is of bounded local treewidth.
Proof. The proof follows from Theorem 6.1. The apex graph Ai, has diameter < 2 and
treewidth > i. So a minor-closed family of graphs with the parameter-treewidth property for P cannot contain all apex graphs and hence it is of bounded local treewidth.
Typical examples of graph parameters satisfying Theorem 7.6 and Lemma 7.7
are dominating set and its generalization q-dominating set, for a fixed constant q (in
149
which each vertex can dominate its q-neighborhood). These graph parameters are
O(r2 )-contraction-bidimensional and their value is I for any apex graph Ai, i > 1.
We can strengthen the "if and only if" result provided by Theorem 7.6 and
Lemma 7.7 with the following lemma. We just need to use the fact that if the
value of P is less than the value of P' then the parameter-treewidth property for P
implies the parameter-treewidth property for P' as well.
Lemma 7.8. Let P be a graph parameter whose value is lower bounded by some
contraction-bidimensionalparameter and let P(Ai) = 0(1) for any i > 1. Then a
minor-closedgraph class F has the parameter-treewidthproperty for P if and only if
F is of bounded local treewidth.
Proof. The "only if" direction is the same as in Lemma 7.7. Suppose now that P'
is a contraction-bidimensional parameter where, for any graph G, P'(G) < P(G).
Applying Theorem 7.6 to P' we obtain that, if F is of bounded local treewidth,
then F has the parameter-treewidth property for P', which means that there exists
a strictly increasing function t such that, for any graph G E F, tw(G) < t(P'(G)).
As P'(G) < P(G), we have that tw(G) < t(P(G)) and thus F has the parametertreewidth property for P.
o
Lemma 7.8 can be used not only for contraction-bidimensional graph parameters.
As an example we mention the clique-transversal number of a graph, i.e., the minimum
number of vertices meeting all the maximal cliques of a graph. (The clique-transversal
number is not contraction-bidimensional because an edge contraction may create a
new maximal clique and the value of the clique-transversal number may increase.) It
is easy to see that this graph parameter always exceeds the domination number (the
size of a minimum dominating set) and that any graph in Ai has a clique-transversal
set of size 1.
Another application is the H-domination number, i.e., the minimum cardinality of
a vertex set that is a dominating set of G and satisfies some property H in G. If this
property is satisfied for any one-element subset of V(G) then we call it regular. Examples of known variants of the parameterized dominating-set problem corresponding
150
to the I-domination
number for some regular property H are the following parame-
terized problems: the independent dominating set problem, the total dominating set
problem, the perfect dominating set problem, and the perfect independent dominating
set problem (see the exact definitions in [2]).
We summarize the previous observations with the following:
Corollary 7.9. Let P be any of the following graph parameters: the minimum car-
dinality of a dominatingset, the minimum cardinalityof a q-dominating set (for any
fixed q), the minimum cardinality of a clique-transversal set, or the minimum cardi-
nality of a dominating set with some regularproperty I. A minor-closedfamily of
graphs F has the parameter-treewidthproperty for P if and only if F is of bounded
local treewidth. The function t(k) in the parameter-treewidth property is 20(Alogk).
7.4
Algorithmic Consequences and Concluding Re-
marks
Courcelle [61] proved a meta-theorem on graphs of bounded treewidth; he showed
that, if
X
is a property of graphs that is definable in monadic second-order logic, then
0 can be decided in linear time on graphs of bounded treewidth. Frick and Grohe [96]
partially extended this result to graphs of bounded local treewidth; they showed that,
for any property 0 that is definable in first-order logic and for any class of graphs
of bounded local treewidth, there is an O(nl+E)-time algorithm deciding whether a
given graph has property A, for any
> 0. The constant in the O notation depends
on 1/E, , and the local treewidth bound. However, the running time of Frick and
Grohe's algorithm remains unanalyzed in terms of 0: their algorithm transforms
into so-called "Gaifman normal form" [98] and the complexity of this transformation
is unknown.
Using Theorems 7.3 and 7.6, we obtain a result along similar lines of Frick and
Grohe. Specifically, consider any property that is solvable in polynomial time on
graphs of bounded treewidth, e.g., properties definable in monadic second-order logic.
151
If the property is minor-bidimensional, we obtain a fixed-parameter algorithm on
general minor-closed graph families excluding some fixed graphs; and if the property
is contraction-bidimensional, we obtain a fixed-parameter algorithm on minor-closed
graph families of bounded local treewidth. The differences between our result and
Frick and Grohe's result are that our properties must be bidimensional but need not
be definable in first-order logic, and our graph families must be minor-closed but
need not have bounded local treewidth (for minor-bidimensional properties). Also,
in contrast to the work of Frick and Grohe, the running time of our algorithm has an
explicit bound.
Theorem 7.10. Let P be a graph parameter such that, given a tree decomposition of
width at most w for a graph G, the graph parameter can be computed in h(w)n
°( l)
time. Now, if P is a g-minor-bidimensionalparameter and G belongsto a minorclosedgraphfamily excludingsomefixed graphs,or P is a g-contraction-bidimensional
parameterand G belongsto a minor-closedfamily of graphsof boundedlocaltreewidth,
then we can computeP on G in h( 2 0(9g -
(k) 1ogg-l(k)))nO(1)220(9-l(k)
logg-l(k))n3+e
time,
for any E > 0.
(k)logg (k))
Proof. The algorithm is as follows. First we check whether tw(G) is in 2 0(9g
By Theorems 7.3 and 7.6, if it is not, graph parameter P has value more than k on
graph G. This step can be performed by Amir's algorithm [11], which for a given
graph G and integer w, either reports that the treewidth of G is at least w, or produces a tree decomposition of width at most (3 + 2)w in time 0(2 3
69&n 3 w3
log4 n).
Thus by using Amir's algorithm we can either compute a tree decomposition of G of
size 2 0( g- (k)lgg'
-
(k))
in time
-(k)logg2 2°(9
(k))n3+e
or conclude that the treewidth of
G is not in 2 0(g-l(k)logg-l(k))
Now if we find a tree decomposition of the aforementioned width, we can compute
P on G in time h(20(g-l(k)logg -(k)))n()
time. The running time of this algorithm is
the one mentioned in the statement of the theorem.
l
For example, let G be a graph from a minor-closed family F of bounded local
treewidth. Because the dominating set of a graph with a given tree decomposition
152
of width at most w can be computed in time 0(2 2 wn) [2], Theorem 7.10 gives an
algorithm which either computes a dominating set of size at most k, or concludes
that there is no such a dominating set in 2 20(V'logk()n(
time. The same result holds
also for computing the minimum size of a q-dominating set. Indeed, Theorem 7.10
can be applied because the q-dominating set of a graph with a given tree decomposition of width at most w can be computed in time O(qo(w)n) [64]. Also, algorithms
on graphs of bounded treewidth for clique-transversal set, and H-domination set appeared in [50] and [2]respectively. Using these algorithms, and the fact that all these
graph parameters are lower bounded by the domination number, the methodology of
the proof of Theorem 7.10 can give algorithmic results for clique-transversal set and
Il-domination set with the same running times as in the case of dominating set (i.e.,
22(/klog
k)nO ))
Finally, it is known that the dominating set problem admits a linear size kernel on
planar graphs [4]. Recently, this result was extended to graphs of bounded genus [94].
It is tempting to ask whether such a kernel exists for any minor-closed graph class
of bounded local treewidth, i.e., any minor-closed graph class with the parametertreewidth property for dominating set. The same question can be asked for other
bidimensional parameters. In particular, we wonder whether there is any link between
linear kernels and bidimensionality.
153
154
Chapter 8
Graphs Excluding a Fixed Minor
have Grids Almost as Large as
Treewidth
The r x r grid graphis the canonical planar graph of treewidth e(r).
In particular,
an important result of Robertson, Seymour, and Thomas [151] is that every planar
graph of treewidth w has an Q(w) x Q(w) grid graph as a minor. Thus every planar
graph of large treewidth has a grid minor certifying that its treewidth is almost as
large (up to constant factors).
In their Graph Minor Theory, Robertson and Seymour [144] have generalized this
result in some sense to any graph excluding a fixed minor: for every graph H and
integer r > 0, there is an integer w > 0 such that every H-minor-free graph with
treewidth at least w has an r x r grid graph as a minor. This result has been reproved by Robertson, Seymour, and Thomas [151], Reed [141], and Diestel, Jensen,
Gorbunov, and Thomassen [80]. The best known bound on w in terms of r is as
follows:
Theorem 8.1. [151,Theorem 5.8] Every H-minor-free graphof treewidthlargerthan
2 0 51v(H)13r
has an r x r grid as a minor.
While the existence of such a relationship between treewidth and grid minors is
155
interesting, this bound of w = 20(r) is much weaker than the bound of w = 0(r)
attainable for the special case of planar graphs. In particular, the grid they obtain
from this theorem can have treewidth logarithmic in the treewidth of the original
graph, which does not serve as much of a certificate of large treewidth as we have for
planar graphs. The main result of this chapter is the following much tighter bound:
Theorem 8.2. For any fixed graph H, every H-minor-free graph of treewidth w has
an Q(w) x Q(w) grid as a minor.
Thus the r x r grid is the canonical H-minor-free graph of treewidth
E)(r)for
any fixed graph H. This result is best possible up to constant factors. Chapter
10 discusses the dependence of the constant factor in the Q notation on the fixed
graph H.
Our result cannot be generalized to arbitrary graphs: Robertson, Seymour, and
Thomas [151]proved that some graphs have treewidth Q(r2 lg r) but have grid minors
only of size O(r) x O(r). The best known relation for general graphs is that having
treewidth more than 202r5 implies the existence of an r x r grid minor [151]. The best
possible bound is believed to be closer to e(r 2 g r) than
2
(r5), perhaps even equal
to O(r 2 lgr) [151].
Our approach in the proof of Theorem 8.2 can be viewed more generally as a
framework for extending combinatorial results on planar graphs to hold on H-minorfree graphs for any fixed H. The framework follows three main steps: extension from
planar graphs to bounded-genus graphs, further extension to "almost-embeddable
graphs", and further extension to clique sums of almost-embeddable graphs. Recall
that almost-embeddable graphs are bounded-genus graphs except for a bounded number of "local areas of non-planarity", called vortices, and for a bounded number of
"apex" vertices, which can have any number of incident edges that are not properly
embedded. The underpinnings of this framework is the structural characterization of
H-minor-free graphs in the Robertson-Seymour Graph Minor Theory [149]. Recently
this framework has been used to generalize many efficient algorithms from planar
graphs to H-minor-free
graphs [63, 103] (see also Chapter
156
5). Our work shows how
the framework can be applied to combinatorial results.
In addition to giving a tight bound on this basic combinatorial problem relating treewidth and grids, our result has many combinatorial consequences, each with
several algorithmic consequences. For instance, one of the main consequences of our
result gives the tightest possible parameter-treewidth
bound for all bidimensional
parameters in all possible H-minor-free graphs:
Theorem 8.3. For any minor-bidimensionalparameter P which is at least g(r) in
the r x r grid, every H-minor-free graph G has treewidth tw(G) = O(g-l(P(G))).
For any contraction-bidimensionalparameterP which is at leastg(r) in an augmented
r x r grid, every apex-minor-freegraphG has treewidthtw(G) = O(g-1(P(G))). In
particular, if g(r) = e(r2 ), then these boundsbecometw(G) = O(P(G).
The proof of this theorem is identical to the proofs of Theorems 7.3 (for minorbidimensional parameters) and 7.6 (for contraction-bidimensional parameters) except
that we substitute the application of Theorem 8.1 with Theorem 8.2.
The reader is referred to Sections 1.6-1.11 to see other concequences of Theorems 8.2 and 8.3 in the bidimensionality theory.
8.1
Overview of Proof of Main Theorem
The proof of our main theorem (Theorem 8.2) is based on a series of reductions. Each
reduction converts a given graph into a "simpler" graph whose treewidth is Q(tw(G)).
The first reduction applies Theorem 1.3 to the original graph G, decomposing it
into a clique sum of almost-embeddable graphs. By Lemma 2.4, at least one summand
in this clique sum has treewidth at least tw(G). Therefore we can focus on this single
summand of large treewidth.
However, we note that this summand may not be a
minor of G, and therefore it is not enough to prove that the summand has a large
grid as a minor; we must deal with this issue later in the proof.
The second, trivial reduction is to remove the apices from the almost-embeddable
graph. This reduction changes the treewidth by at most an additive constant. Now
our almost-embeddable graph is apex-free.
157
The third reduction effectively removes the vortices from the apex-free almostembeddable graph. This reduction uses that vortices have small pathwidth to conclude that the treewidth remains roughly the same. At this point the graph has
bounded genus, because we have removed both apices and vortices.
Because the graph has bounded genus, it has a large grid as a minor. However,
this grid is not useful: the graph is not necessarily a minor of the original graph G
because, during the clique-sum decomposition, we may have introduced extra edges
when the join set was completed into a clique. We call such edges virtual edges, and all
other edges actual edges. One difficulty of Theorem 1.3 is that it does not guarantee
that the virtual edges can be obtained by taking a minor of the original graph G,
and therefore the pieces may not be minors of G. The fourth reduction overcomes
this difficulty by obtaining some virtual edges by taking minors of the original graph
G, and removes other virtual edges which cannot be obtained, while still preserving
the treewidth up to constant factors. We call the resulting graph an approximation
graph.
The approximation graph is both a minor of G and has bounded genus. Now we
use the fact that a bounded-genus graph with treewidth w has an Q(w) x Q(w) grid
as a minor. Therefore both the approximation graph and G have such a grid as a
minor.
8.2
Proof of Main Theorem
In this section we prove Theorem 8.2.
First, we will need the following property about how treewidth changes during
small operations to faces of a graph:
Lemma 8.4. Consider any graph G embeddedin some surface of genus g, with
tw(G) =
(g2 ). If G' is the result of contracting a face of G to a point, then
tw(G') < tw(G) and tw(G') = Q(tw(G)/(g + 1)).
Proof. Let f denote the face of G contracted to form G'. Because G' is a minor of G,
tw(G')
tw(G).
Consider the graph G" formed from graph G by adding a new
158
vertex v in the middle of face f and adding an edge connecting v to every vertex
of f. This graph G" is embedded in the same genus-g surface as G. The treewidth
of G" is at most 1 larger than the treewidth of G, by adding v to all bags of a tree
decomposition of G. By [73, Theorem 2], there is a sequence of contractions that
brings G" to a graph R that is a (planar) partially triangulated r x r grid augmented
with at most g additional edges, where r = Q(tw(G")/(g + 1)) = Q(tw(G)/(g + 1)).
Every vertex in R can be labeled by the set of vertices in G" that were contracted to
form it. Let
R
denote the vertex in R whose label includes v. For every neighbor
w of v in G, the vertex
vR
WR
in R whose label includes w has distance at most 1 from
in R, because contractions only decrease distances. We modify the augmented
partially triangulated grid R as follows. For every neighbor w of v in G for which
WR
VR,
the edge
we delete all edges incident to
{VR, WR}.
WR
except (vR,
WR},
and then we contract
The resulting graph R' is a minor of R and thus of G". If we
re-order the sequence of contractions and deletions that bring G" to R' to start with
the contractions of the edges between v and the vertices of face f (which is equivalent
to contracting the face f in the original graph G), then the succeeding sequence
of contractions and deletions brings G' to R'. Therefore R' is a minor of G'. By
ignoring every row or column of the grid in which an edge was deleted, we obtain an
(r - g -4) x (r - g - 4) grid minor of G'. (There may be g such rows (resp., columns)
from the g additional edges, 2 from the neighborhood of v, and 2 from the boundary
of the grid.) Therefore tw(G') > r - g - 5 = Q(tw(G)/(g + 1)).
l
Now we apply Theorem 1.3 to the original graph G, decomposing it into a clique
sum of almost-embeddable graphs.
Lemma 8.5. At least one summand in the cliquesum has treewidth at least tw(G).
Proof. Immediate by Lemma 2.4.
l
Let G' denote a summand in the clique sum with tw(G') > tw(G).
For every
vertex v in G', there is a corresponding vertex f(v) in G by following the definition
of clique sum. Each edge {u, v} in G' may or may not have a corresponding edge
159
(f(u), f(v)} in G. If the edge {f(u), f(v)} exists in G, we say that {u, v} is an actual
edge in G'; otherwise, it is a virtual edge in G'. Virtual edges arise from removing
edges from the join set during a clique sum.
Because G' is h-almost-embeddable in some bounded-genus surface, it consists of
a bounded-genus graph augmented by at most h vortices and at most h apices. We
remove all apices from G' to produce an apex-free h-almost-embeddable graph G".
Because adding a vertex and any collection of incident edges to a graph can increase
the treewidth by at most 1, we have the following relation between the treewidths of
G' and G":
Lemma 8.6. tw(G") > tw(G')- h.
Next we remove all vortices from G". Let G' denote the bounded-genus part of
the apex-free h-almost-embeddable graph G", and let Ui denote the set of vertices at
which vortex i is attached to Go (as in definition of h-almost embeddable graphs).
Define G"' = Go - U1
U2 -
-
Uh, i.e., G"' is the result of removing all vertices
from vortices in G".
Lemma 8.7. tw(G"') = O(tw(G")) for h = 0(1).
Proof. Suppose G" decomposes into Gg U G'LU G2'U ... U G where each G' i > 1,
is a vortex as in definition of h-almost embeddable graphs. Define an intermediate
graph G as follows. Let U = {ul, u,.. ., u}mi}be the cyclically ordered vertices of Go
at
(~C,
iwhich
WI1~11, vortex
VIC~C GI'
i is attached. We obtain G by starting from G' and adding edges
{u, u2+l} where they do not already exist, and where j +1 is treated modulo mi, for
each 1 < i < h and each 1 < j < mi. Because we only added a planar graph within
the face corresponding to Ui, G is embeddable in the same bounded-genus surface
as G.
We claim that tw(G") < (h + 1)2 (tw(G) + 1). Consider some minimum-width
tree decomposition of
, and consider each bag B of that tree decomposition. For
each u that occurs in bag B, we add to B the corresponding bag 13a from the path
decomposition of vortex G'.
because
The resulting bags form a tree decomposition of G"
u l , ui,.. . , umi are connected in a path in G. By charging the < h + 1
160
added vertices to the occurrence of uJ that triggered the addition, each bag increases
in size by a factor at most h + 1 for each of the h vortices. Thus the width of this
tree decomposition of G" is at most (h(h-+ 1)) (tw(G) + 1)- 1, which is stronger than
the desired claim.
Let G be the graph resulting from G by contracting the face {u, u,...,
u
)
in G into a single vertex, for each i. Applying Lemma 8.4, h times, we obtain
tw(G) = Q(tw(G)) because h and the genus of the surface in which G is embedded
are 0(1). Therefore tw(G") = O(tw(G)).
Finally we delete each contracted vertex in G, which results in G"'. Thus tw(G"') >
tw(G) - h, so tw(G") = O(tw(G"')) as desired.
[
At this point the graph has bounded genus, because we have removed both apices
and vortices. In the next step we deal with virtual edges. Intuitively, for each summand G' in the clique-sum decomposition of the original graph G, we construct a
graph G which is a minor of G and "approximately" preserves the virtual edges within
G'. For this step we need an additional property of the clique-sum decomposition obtained in the proof of Theorem 1.3: each clique sum involves at most three vertices
from each summand other than apices and vertices in vortices of that summand. This
stronger form of Theorem 1.3 follows from exactly the same proof from [149]. At a
high level, the proof of Theorem 1.3 ([149, Theorem 1.3]) consists of two components,
[149, Theorem 2.3] and [149, Theorem 3.1]. The first part uses clique sums that involve only apices and vertices in vortices, while the second part uses clique sums that
involve only three vertices in each summand (from the 3-separations). When these
clique sums are combined, we may obtain clique sums involving apices, vertices in
vortices, and up to three additional vertices in each summand. This fact has been
confirmed independently by Seymour [154].
Definition
8.8. Let GC'be an h-almost-embeddable graph in a clique-sum decom-
position of a graph G arising from Theorem 1.3. The approximation graph of G',
denoted by G, is obtained by starting from G"', removing the virtual edges, and replacing some of them as follows. In the clique-sum decomposition of G, for each clique
161
sum involving G' with the property that the join set W has IW n V(G"')I > 1, we do
the following:
1. If IW n V(G"')I = 2, we add an edge between these two vertices.
2. If IW n V(G"')J = 3 and there is more than one clique sum that contains W n
V(G"') in its join set, we add all edgesbetweenpairs of vertices in W n V(G"').
3. If W n V(G')I = 3 and there is only one clique sum that containsW n V(G"')
in its join set, we add a new vertex v inside the triangle of W n V(G"') on the
surface and then add an edge connecting v to each vertex of W n V(G"').
Lemma 8.9. Let G' be an h-almost-embeddablegraphin a clique-sumdecomposition
of a graph G arising from Theorem 1.3. The approximation graph G of G' is a minor
of G and can be embedded in the same surface as the bounded-genus part of G'.
Proof. First, G'" with all virtual edges removed is a minor of G, because the former
graph can be constructed from G by deleting all vertices not in the summand G' and
deleting all apices and vertices in vortices in G'. All that remains to show is that
the edges added in Cases 1-3 of Definition 8.8 can also be formed as a minor of G.
We use the (trivial) additional property of the clique-sum decomposition arising from
Theorem 1.3 that each summand in the clique sum is connected even after removal
of the join set. (If a summand were not connected after the removal of the join set,
we could rewrite the initial clique-sum decomposition by splitting the summand into
a clique sum of these pieces.) Now, for each clique sum between G' and F with the
property that the join set W has W n V(G')I > 1, we contract F down to a single
vertex v adjacent to all vertices in the join set. In Case 3, this vertex v is precisely
the desired vertex v inside the triangle W n V(G"'). This triangle is guaranteed to
be empty in the bounded-genus part of G' in the clique-sum decomposition arising
from Theorem 1.3; if this were not the case, again we could rewrite the clique-sum
decomposition by splitting G' into a clique sum of two pieces. Thus the resulting
graph can be embedded in the same surface as the bounded-genus part of G'. In the
other two cases, we contract v into a vertex of W n V(G"')-in
162
Case 2, we contract
two different v's into two different vertices of W n V(G"')-and
obtain the additional
edges added to G. Finally, we delete the apices and vertices in vortices in G', and
delete any other summands that had IWnV(G"')I < 1. In the end we have contracted
and deleted edges in G to obtain precisely G.
D
Lemma 8.10. tw(G) > (tw(G"') + 1)-1.
Proof. To prove that tw(G"') < 3(tw(G) + 1) - 1, we start from a minimum-width
tree decomposition of 0 and convert it into a tree decomposition of G " '. We need only
consider Case 3 in Definition 8.8 because otherwise G is identical to G"'. For each
occurrence of an added vertex v from Case 3 in a bag B in the tree decomposition of
G, we replace v in B with all three vertices from W n V(G"'). The result is a tree
decomposition of G"' where each bag has increased in size by at most a factor of 3.
By Lemma 8.9, the approximation graph G is both a minor of G and has bounded
genus. By [63, Theorem 3.5] (see also Theorem 5.14), every bounded-genus graph
with treewidth Q(r) has an r x r grid as a minor. By Lemmas 8.5, 8.6, 8.7, and 8.10,
tw(G) = Q(tw(G)). Therefore G and thus G have an Q(tw(G)) x Q(tw(G)) grid as
a minor. This concludes the proof of Theorem 8.2.
163
164
Chapter 9
Improved Approximation
Algorithms for Minimum-Weight
Vertex Separators and Treewidth
Given a graph G = (V, E), one is often interested in finding a small "separator" whose
removal from the graph leaves two sets of vertices of roughly equal size (say, of size at
most 21VI/3), with no edges connecting these two sets. The separator itself may be
a set of edges, in which case it is called a balanced edge separator, or a set of vertices,
in which case it is called a balanced vertex separator. In the present work, we focus
on vertex separators.
Balanced separators of small size are important in several contexts. They are
often the bottlenecks in communication networks (when the graph represents such a
network), and can be used in order to provide lower bounds on communication tasks
(see e.g.
[130, 128, 28]). Perhaps more importantly, finding balanced separators of
small size is a major primitive for many graph algorithms, and in particular, for those
that are based on the divide and conquer paradigm [132, 28, 129].
Certain families of graphs always have small vertex separators.
trees always have a vertex separator containing a single vertex.
For example,
The well known
planar separator theorem of Lipton and Tarjan [132]shows that every n-vertex planar
graph has a balanced vertex separator of size O(vJn), and moreover, that such a
165
separator can be found in polynomial time. This was later extended to show that
more general families of graphs (any family of graphs that excludes some minor, and
certain geometric graphs) have small separators [101, 10, 135]. However, there are
families of graphs (for example, expander graphs and the complete graph) in which
the smallest separator is of size Q(n).
Finding the smallest separator is an NP-hard problem (see, e.g. [45]). In this
chapter, we study approximation algorithms that find vertex separators whose size is
not much larger than the optimal separator of the input graph. These algorithms can
be useful in detecting small separators in graphs that happen to have small separators,
as well as in demonstrating that an input graph does not have any small vertex
separator (and hence, for example, does not have serious bottlenecks for routing).
Much of the previous work on approximating vertex separators piggy-backed on
work for approximating edge separators. For graphs of bounded degree, the sizes of
the minimum edge and vertex separators are the same up to a constant multiplicative factor, leading to a corresponding similarity in terms of approximation ratios.
However, for general graphs (with no bound on the degree), the situation is different.
For example, every edge separator for the star graph has Q(n) edges, whereas the
minimum vertex separator has just one vertex. There are simple reductions from
the problem of approximating edge separators to the the problem of approximating
vertex separators. (For example, replace every vertex v by clique Cv on n 3 vertices,
and every original edge e = (u, v) by a vertex ve connected to all vertices of Cu and
Cv.) As to the reverse direction, it is only known how to reduce the problem of
approximating vertex separators to the problem of approximating edge separators in
directed graphs (a notion that will not be discussed in this chapter).
The previous best approximation ratio for vertex separators is based on the work
of Leighton and Rao [129]. They presented an algorithm based on linear programming that approximates the minimum edge separator within a ratio of O(log n). They
observed that their algorithm can be extended to work on directed graphs, and hence
gives an approximation ratio of O(logn) also for vertex separators, using the algorithm for (directed) edge separators as a black box. More recently, Arora, Rao and
166
Vazirani [19] presented an improved algorithm based on semidefinite programming
that approximates the minimum edge separator within a ratio of O(Vlo/-n).
Their
remarkable techniques, which are a principal component in our algorithm for vertex
separators, are discussed more in the following section.
In the present work, we formulate new linear and semidefinite program relaxations
for the vertex separator problem, and then develop rounding algorithms for these
programs. The rounding algorithms are based on techniques that were developed in
the context of edge separators, but we exploit new properties of these techniques and
adapt and enhance them to the case of vertex separators. Using this approach, we
improve the best approximation ratio for vertex separators to O( log/n). In fact,
one can obtain an O(V/logopt) approximation, where opt is the size of an optimal
separator (see [90]). (An O(log opt) approximation can be derived from the techniques
of [129].) In addition, we derive and extend some previously known results in a unified
way, such as a constant factor approximation for vertex separators in planar graphs
(a result originally proved in [12]), which we extend to any family of graphs excluding
a fixed minor.
Before delving into more details, let us mention two aspects in which edge and
vertex separators differ. One is the notion of a minimum ratio cut, which is an
important notion used in our analysis. For edge cuts, all "natural" definitions of such
a notion are essentially equivalent. For vertex separators, this is not the case. One
consequence of this is that our algorithms provide a good approximation for vertex
expansion in bounded degree graphs, but not in general graphs. This issue will be
discussed in Section 9.3. Another aspect where there is a distinction between edge
and vertex separators is that of the role of embeddings into L 1 (a term that will be
discussed later). For edge separators, if the linear or semidefinite program relaxations
happen to provide such an embedding (i.e. if the solution is an L 1 metric), then they in
fact yield an optimal edge separator. For vertex separators, embeddings into L1 seem
to be insufficient, and we give a number of examples that demonstrate this deficiency.
Our rounding techniques for the vertex separator case are based on embeddings with
small average distortion into a line, a concept that was first systematically studied
167
by Rabinovich [139].
As mentioned above, finding small vertex separators is a basic primitive that is
used in many graph algorithms. Consequently, our improved approximation algorithm for minimum vertex separators can be plugged into many of these algorithms,
improving either the quality of the solution that they produce, or their running time.
Rather than attempting to provide in this chapter a survey of all potential applications, we shall present one major application, that of improving the approximation
ratio for treewidth, and discuss some of its consequences.
9.1
Related Work
An important concept that we shall use is the ratio of a vertex separator (A, B, S).
Given a weight function r : V - R+ on vertices, and a set S C V which separates G
into two disconnected pieces A and B, we can define the sparsity of the separator by
7r(S)
min{7r(A),r(B)} + r(S)'
Indeed, most of our effort will focus on finding separators (A, B, S) for which the
sparsity is close to minimal among all vertex separators in G.
In the case of edge separators, there are intimate connections between approximation algorithms for minimum ratio cuts and the theory of metric embeddings. In
particular, Aumann and Rabani [21] and Linial, London, and Rabinovich [131]show
that one can use L1 embeddings to round the solution to a linear relaxation of the
problem. For the case of vertex cuts, we will show that L 1 embeddings (and even
Euclidean embeddings) are insufficient, but that the additional structure provided by
many embedding theorems does suffice. This structure corresponds to the fact that
many embeddings are of Frchet-type, i.e. their basic component takes a metric space
X and a subset A C X and maps every point x E X to its distance to A. This
includes, for instance, the classical theorem of Bourgain [43].
The seminal work of Leighton and Rao [129] showed that, in both the edge and
168
vertex case, one can achieve an O(log n) approximation algorithm for minimum-ratio
cuts, based on a linear relaxation of the problem. Their solution also yields the first
approximate max-flow/min-cut theorems in a model with uniform demands.
The
papers [131, 21] extend their techniques for the edge case to non-uniform demands.
Their main tool is Bourgain's theorem [43], which states that every n-point metric
space embeds into L 1 with O(log n) distortion.
Recently, Arora, Rao, and Vazirani [19] exhibit an O(Vl/ogn) approximation for
finding minimum-ratio edge cuts, based on a known semi-definite relaxation of the
problem, and a fundamentally new technique for exploiting the global structure of the
solution. This approach, combined with the embedding technique of Krauthgamer,
Lee, Mendel, and Naor [124], has been extended further to obtain approximation
algorithms for minimum-ratio edge cuts with non-uniform demands. In particular,
by combining [19], [124], and the quantitative improvements of Lee [126], Chawla,
Gupta, and Racke [52] exhibit an O(logn)
3/4
approximation. More recently, Arora,
Lee, and Naor [18] have improved this bound almost to that of the uniform case,
yielding an approximation ratio of O(V/ogn log log n).
On the other hand, progress on the vertex case has been significantly slower. In
the sections that follow, we attempt to close this gap by providing new techniques for
finding approximately optimal vertex separators.
9.2
Results and Techniques
In Section 9.3, we introduce a new semi-definite relaxation for the problem of finding minimum-ratio vertex cuts in a general graph. In preparation for applying the
techniques of [19], the relaxation includes so-called "triangle inequality" constraints
on the variables. The program contains strictly more than one variable per vertex of
the graph, but the SDP is constructed carefully to lead to a single metric of negative
type 1 on the vertices which contains all the information necessary to perform the
1
A metric space (X,d) is said to be of negative type if d(x,y) = [If(x)
f :X - L2.
169
-
f(y)112 for some map
rounding.
In Section 9.4, we exhibit a general technique for rounding the solution to optimization problems involving "fractional" vertex cuts. These are based on the ability
to find line embeddings with small average distortion, as defined by Rabinovich [139]
(though we extend his definition to allow for arbitrary weights in the average). In
[139], it is proved that one can obtain constant factor average distortion embeddings
into the line for metrics supported on planar graphs. This is observed only as an interesting structural fact, without additional algorithmic consequences over the known
average distortion embeddings into all of L 1 [140, 120]. For the vertex case, we will
see that this additional structure is crucial.
Using the seminal results of [19], which can be viewed as a line embedding, we
then show that the solution of the semi-definite relaxation can be rounded to a vertex
separator whose ratio is within O(v/Ign)
of the optimal separator. In the standard
SDP for minimum-ratio edge cuts (employed in the algorithms of [19]), no lower bound
is known on the integrality gap. Very recent work of Khot and Vishnoi [119] shows
that in the non-uniformdemand case, the gap must tend to infinity with the sizeof the
instance. In contrast, we show that our analysis is tight by exhibiting an Q(/lo-in)
integrality gap for the SDP in Section 9.6. Interestingly, this gap is achieved by an
L1 metric. This shows that L1 metrics are not as intimately connected to vertex cuts
as they are to edge cuts, and that the use of the structural theorems discussed in the
previous paragraph is crucial to obtaining an improved bound.
We exhibit an O(log k)-approximate max-flow/min-vertex-cut theorems for general instances with k commodities. The best previous bound of O(log3 k) is due to
[88] (they actually show this bound for directed instances with symmetric demands,
but this implies the vertex case). This is proved in Section 9.5. A well-known reduction shows that this theorem implies the edge version of [131, 21] as a special case.
Again, our rounding makes use of the general tools developed in Section 9.4 based
on average-distortion line embeddings. In Section 9.5.2 we show that any approach
based on L 1 embeddings and Euclidean embeddings, respectively, must fail since the
integrality gap can be very large even for such metric spaces. Using the improved line
170
embeddings for metrics on graphs which exclude a fixed minor [139] (based on [120]
and [140]), we also achieve a constant-factor approximation for finding minimum ratio
vertex cuts in these families. This answers an open problem asked in [71].
By improving the approximation ratios for balanced vertex separators in general
graphs and graphs excluding a fixed minor, we improve the approximation factors for
a number of problems relating to graph-theoretic decompositions such as treewidth,
branchwidth, and pathwidth. For instance, we show that in any graph of treewidth
k, we can find a tree decomposition of width at most O(kl/`-k).
If the input graph
excludes some fixed minor, we give an algorithm that finds a decomposition of width
O(k). A discussion of these problems, along with the salient definitions, appears in
Section 9.7. See Theorem 9.16 and Corollary 9.17 for a list of the problems to which
our techniques apply.
Improving the approximation factor for treewidth in general graphs and graphs
excluding a fixed minor to O(V/ogni) and O(1), respectively, answers an open problem
of [71], and leads to an improvement in the running time of approximation schemes
and sub-exponential fixed-parameter algorithms for several NP-hard problems on
graphs excluding a fixed minor. For instance, we obtain the first polynomial-time
approximation schemes (PTAS) for problems like minimum feedback vertex set and
connected dominating set in such graphs (see Theorem 9.18 for a full list). Finally, our
techniques yield an O(g)-approximation algorithm for the vertex separator problem
in graphs of genus at most g. It is known that such graphs have balanced separators
of size O(V/g-) [101], and that these separators can be found efficiently [117] (earlier, [10] gave a more general algorithm which, in particular, finds separators of size
0(
3
/
2 n)).
Our approximation algorithms thus finds separators of size O(
g/n),
but when the graph at hand has a smaller separator, our algorithms perform much
better than the worst-case bounds of [101, 10, 117].
171
9.3
A Vector Program for Minimum-Ratio Vertex
Cuts
Let G = (V, E) be a graph with non-negative vertex weights 7r: V -- [0, oo). For a
subset U C V, we write r(U) = EuEU 7r(u). A vertex separator partitions the graph
into three parts, S (the set of vertices in the separator), A and B (the two parts that
are separated). We use the convention that 7r(A) > r(B). We are interested in finding
separators that minimize the ratio of the "cost" of the separator to its "benefit." Here,
the cost of a separator is simply 7r(S). As to the benefit of a separator, it turns out
that there is more than one natural way in which one can define it. The distinction
between the various definitions is relatively unimportant whenever r(S) < r(B), but
it becomes significant when 7r(S) > 7r(B). We elaborate on this below.
In analogy to the case of edge separators, one may wish to take the benefit to be
r(B). Then we would like to find a separator that minimizes the ratio r(S)/(B).
However, there is evidence that no polynomial time algorithm can achieve an approximation ratio of O(1V16 ) for this problem (for some 6 > 0). See [90] for details.
For the use of separators in divide and conquer algorithms, the benefit is in the
reduction in size of the parts that remain. This reduction is ir(B) + ir(S) rather than
just Ir(B), and the quality of a separator is defined to be
r(S)
7r(B) + 7r(S)
This definition is used in the introduction of this Chapter, and in some other earlier
work (e.g. [12]).
As a matter of convenience, we use a slightly different definition. We shall define
the sparsity of a separator to be
ca(A, B, S)=
7r(S
777(AU S) 7r(B U S)
Under our convention that r(A) > 7r(B),we have that 7r(V)/2 < 7r(AUS)< 7r(V),
and the two definitions differ by a factor of
172
E(7r(V)).
We define ar(G) to be the minimum over all vertex separators (A, B, S) of a,(A, B, S).
The problem of computing a,(G) is NP-hard (see [45]). Our goal is to give algorithms for finding separators (A, B, S) for which ar(A, B, S)
O(,l\/&)
aa,(G),
where k = supp(r)l is the number of vertices with non-zero weight in G.
Before we move onto the main algorithm, let us define
a (A, B, S) = 7r(S)/[w(A)- r(B U S)].
Note that ar(A, B, S) and oa(A, B, S) are equivalent up to a factor of 2 whenever
7r(A) > 7(S). Hence in this case it will suffice to find a separator (A, B, S) with
, (A, B, S) < O(/og k) d, (G). Allowingourselvesto compare a,7 (A, B, S) to d&(G)
rather than a 7 (G) eases the formulation of the semi-definiterelaxations that follow.
When 7(S) > 7r(A), & no longer provides a good approximation to a. (Moreover, it
is hard to approximate &(G) in this case, as shown in [90].) However, in this case
the trivial separator (0, 0, V) has sparsity at most a constant factor larger than a(G),
making the approximation of a(G) trivial.
9.3.1
The Quadratic Program
We present a quadratic program for the problem of finding min-ratio vertex cuts.
All constraints in this program involve only terms that are quadratic (products of
two variables). Our goal is for the value of the quadratic program to be equal to
&7,(G). Let G = (V, E) be a vertex-weighted graph, and let (A*, B*, S*) be an optimal
separator according to &,(.), i.e. such that d,(G) = -,(A*, B*, S*).
With every vertex i E V, we associate three indicator 0/1 variables, xi, yi and si.
It is our intention that for every vertex exactly one indicator variable will have the
value 1, and that the other two will have value 0. Specifically, xi = 1 if i E A*, Yi = 1
if i E B*, and si = 1 if i E S*. To enforce this, we formulate the following two sets
of constraints.
Exclusion constraints. These force at least two of the indicator variables to be 0.
173
xi ' yi = O, xi si = O, yi si = 0, for every i E V.
Choice constraints. These force the non-zero indicator variable to have value 1.
1, for all i E V.
2
x2
The combination of exclusion and choice constraints imply the following integrality
constraints, which we formulate here for completeness, even though they are not
explicitly included as part of the quadratic program: x i E {0, 1},
E 0, 1}, s E
{0, 1}, for all i E V.
Edge constraints. This set of 2 jIE constraints express the fact that there are no edges
connecting A and B.
xi yj = 0 and xj yi = 0, for all (i,j) E E.
Now we wish to express the fact that we are minimizing d&(A, B, S) over all vertex
separators (A, B, S). To simplify our presentation, it will be convenient to assume
that we know the value K = r(A*) · 7r(B* U S*). This assumption can be made
without loss of generality because it suffices to know the value of K up to a constant
multiplicative factor, and there are at most polynomially many values to choose from
(e.g., guess the heaviest vertex v in A*, and guess which value of 1
i < log n makes
2i7r(v) the closest estimate for 7r(A*)). Alternatively, the assumption can be dropped
at the expense of a more complicated relaxation.
Spreading constraint. The following constraint expresses our guess for the value of K.
E
(i)r(j)(xi - xj)2 = K.
i,jEV
Notice that (xi -
j)2 = 1 if and only if {xi, xj} = {0, 1}.
174
The objective function. Finally, we write down the objective we are trying to minimize:
1
K
minimize
E
iEV
7r(i)s.
The above quadratic program computes exactly the value of & (G), and hence is
NP-hard to solve.
9.3.2
The Vector Relaxation
We relax the quadratic program of Section 9.3.1 to a "vector" program that can be
solved up to arbitrary precision in polynomial time. The relaxation involves two
aspects.
Interpretation
Rd,
of variables.
All variables are allowed to be arbitrary vectors in
rather than in R. The dimension d is not constrained, and might be as large as
the number of variables (i.e., 3n).
Interpretation of products. The original quadratic program involvedproducts
over pairs of variables. Every such product is interpreted as an inner product between
the respective vector variables. The exclusion constraints merely force vectors to be
orthogonal (rather than forcing one of them to be 0), and the integrality constraints
are no longer implied by the exclusion and choice constraints. The choice constraints
imply (among other things) that no vector has norm greater than 1, and the edge
constraints imply that whenever (i, j) E E, the corresponding vectors xi and yj are
orthogonal.
9.3.3
Adding Valid Constraints
We now strengthen the vector program by adding more valid constraints. This should
be done in a way that will not violate feasibility (in cases where the original quadratic
program was feasible), and moreover, that preserves polynomial time solvability (up
to arbitrary precision) of the resulting vector program. It is known that this last
condition is satisfied if we only add constraints that are linear over inner products of
175
pairs of vectors, and this is indeed what we shall do. The reader is encouraged to
check that every constraint that we add is indeed satisfied by feasible 0/1 solutions
to the original quadratic program.
The 1-vector. We add additional variable v to the vector program. It is our intention
that variables whose value is 1 in the quadratic program will have value equal to
that of v in the vector program. Hence v is a unit vector, and we add the constraint
v 2 = 1.
Sphere constraints. For every vector variable z we add the constraint z 2 = vz. Geometrically, this forces all vectors to lie on the surface of a sphere of radius
at v2 because the constraint is equivalent to (z
_
2
)2
2
centered
_ 1
Triangle constraints. For every three variables z 1, z 2 , z3 we add the constraint
(Z1 - Z2)2 + (z2
-
2
3)
(z 1 -
This implies that every three variables (which are points on the sphere S(v, )) form
a triangle whose angles are all at most 7r/2.
Removing the si vectors. In the upcoming sections we shall describe and analyze a
rounding procedure for our vector program. It turns out that our rounding procedure
does not use the vectors si, only the values s = 1- xi-
y2. Hence we can modify
the choice constraints to
x2
a2 + y2 <1
and remove all explicit mention of the si vectors, without affecting our analysis for
the rounding procedure. Similarly, it suffices to include in the vector program only
those triangle constraints in which all three vectors are x vectors. The full vector
program follows.
176
minimize
iEv r(i)(1 -
subject to xi, yi, v
E
2
)
x -
iE V
,
zx+ y2<1,
iEv
Xi Yi = O,
iEV
Xi yj = Xj Yi = 0,
(i,j)
v
2
=
1
V-·X i =X,
2
EE
V· Yi -
,
-i,jeV r(i)r(j)(xi
(Xi - Xj) 2 <
iEV
Xj)-2
(X i- x h ) 2 + (
=
K
X-
j)2, h,i,j E V.
In the following section, we will show how to round this to a solution which is within
an O(Vi-g)
factor of optimal. In Section 9.6, we show that this analysis is tight,
even for a family of stronger (i.e. more constrained) vector programs.
9.4
Algorithmic Framework for Rounding
In this section, we develop a general algorithmic framework for rounding solutions to
optimization problems on vertex cuts. We begin with a classical theorem.
Theorem 9.1 (Menger's theorem). A graph G = (V, E) contains at least k vertexdisjoint paths between two non-adjacent vertices u, v E V if and only if every vertex
cut that separatesu from v has size at least k.
It is well-known that a smallest vertex cut separating u from v can be found in
polynomial time (in the size of G) by deriving Menger's Theorem from the Max-FlowMin-Cut Theorem (see e.g. [165]).
Suppose that, in addition to a graph G = (V, E), we have a set of non-negative
vertex capacities {cv}VEv
C
N. (For simplicity, we are assuming here that capacities
177
are integer, but the following discussion can also be extended to the case of arbitrary
nonnegative capacities.) For a subset S C V, we define cap(S) =
ves
C. We have
the following immediate corollary whose proof is deferred.
Corollary 9.2. Let G = (V, E) be a graph with vertex capacities. Then for any two
non-adjacent vertices u, v E V, the following two statements are equivalent.
1. Every vertex cut S C V that separatesu from v has cap(S) > k.
2. There exist u-v paths P1,P2, ... ,Pk C V such that for every w E V,
#{1 < i < k: w Epi} < c.
Furthermore, a vertex cut S of minimal capacitycan be found in polynomial time.
Proof. The proof is by a simple reduction.
From G = (V, E) and the capacities
{cV}~EV,we create a new uncapacitated instance G' = (V',E')
and then apply
Menger's theorem to G'.
To arrive at G', we replace every vertex v E V with a collection of representatives
v 1, v2 ,..., v
(if c = 0, then this corresponds to deleting v from the graph). Now
for every edge (u, v) E E, we add edges {(ui, vj) : 1 < i < c, 1 < j < c}. It is not
hard to see that every minimal vertex cut either takes all representatives of a vertex
or none, giving a one-to-one correspondence between minimal vertex cuts in G and
o
G'.
Furthermore, given such a capacitated instance G = (V, E), {cV}VEvalong with
u, v E V, it is possible to find, in polynomial time, a vertex cut S C V of minimal
capacity which separates u from v.
Suppose additionally that we have a demand function w : V x V -- R+ which
is symmetric, i.e. w(u, v) = w(v, u). In this case, we define the sparsity of (A, B, S)
with respect to w by
caP,w(A, B, S) =
178AUS
178
capus(
ZvEBUS
W(U, v)V
We definethe sparsity of G by aCaPW(G)
= min{aCaP'W(A,
B, S)} where the minimum
is taken over all vertex separators.
Note that a,(A, B, S) = aP,w(A, B, S) when
Cv = r(v) and w(u, v) = 7r(u) r(v) for all u, v E V.
9.4.1
Line Embeddings and Vertex Separators
Let G = (V, E) be a graph with vertex capacities {cv}v,
w:V x V
-
and a demand function
1R+.Furthermore, suppose that we have a map f : V
R. We give the
following algorithm which computes a vertex cut (A, B, S) in G.
Algorithm FINDCUT(G,
f)
1. Sort the vertices v E V according to the value of f(v): {y, Y2,... , Yn}
2. For each 1 < i < n,
3.
Create the augmented graph Gi = (V U {s, t}, Ei) with
Ei = EU (s, yj), (k, t)': 1 < j < i, i < k < n}.
4.
Find the minimumcapacity s-t separator Si in Gi.
5.
Let A U {s} be the component of G[V U {s, t} \ Si] containing s,
let Bi = V \ (Ai USi).
6. Output the vertex separator (Ai, Bi, Si) for which aCaP'W(Ai,
Bi, Si) is minimal.
The analysis.
the map f
Suppose that we have a cost function cost : V --* R+. We say that
V -
I is path-compatible with the cost function cost if, for any path
=V1,V2,...,vk in G,
k
cost(vi)> If(vl) - f(vk)I.
(9.1)
i=l
We now move onto the main lemma of this section.
Lemma 9.3 (Charging lemma). Let G = (V, E) be any capacitated graph with demand
function w : V x V -- IR+. Suppose additionally that we have a cost function cost :
V - IR+and a path-compatiblemap f : V
179
R. If ao is the sparsity of the minimum
ratio vertex cut output by FINDCUT(G, f), then
E w(u,v)f(u)-
cost(v) >
Zcv
f(v).
,vEV
vEV
Proof. Recall that we have sorted the vertices v according to the value of f(v):
{Y1,Y2,...
,Yn}.
Let Ci = {yl,. .. ,yi} and i = f(Yi+l) - f(Yi). First we have the
following lemma which relates the size of the separators found to the average distance
under f, according to w.
Lemma 9.4.
n-1
E w(u, v)lf(u) - f(v)l.
Ei cap(Si) > ao
i=l
u,vEV
'1
Proof. Using the fact that ao is the minimum sparsity of all cuts found by FINDCUT(G, f),
cap(Si)
E
E
ao
w(u,v)
uEAiUS vEBiUSi
S W(UV).
E
> ao
uEC i vEV\Ci
Multiplying both sides of the previous inequality by eiand summing over i E {1, 2,..., nO
1} proves the lemma.
Now we come to the heart of the charging argument which relates the cost function
to the capacity of the cuts occurring in the algorithm.
Lemma 9.5 (Charging against balls).
n-l
Cv.cost(v)
_>
2
Eicap(Si).
i=1
vEV
Proof. We show how to charge every "unit" of Ei= 1 Eicap(Si) against vertices v e V
so that no vertex is charged more than 2 cv' cost(v). This is done as follows. For every
vertex v E V, we surround f(v) E Il by a closed ball of radius cost(v). This ball has
width 2 cost(v) on the line. We will be charging various amounts to segments of this
width, never charging a segment of v's ball more than c,.
180
Fix some 1
i < n - 1. Since Si is a minimum capacity s-t separator, applying
Menger's theorem yields a family of m = ISil s-t paths
l,. .. ,Pm which use no vertex
v E V more than cv times. Since these paths cross from Ci to V \ Ci, there must exist
subpaths q 1 ,..., qm for which each initial vertex is in Ci and each final vertex is in
V \ Ci.
For each path qj, we will charge Ei against
vEV
cv
C.cost(v).
Let qj
=
Ul, ,..
u2
.
First, we note that since f is path-compatible with cost, the balls {B(ui, cost(ui))}
must coverthe entire interval [f(ul), f(ur)]. Furthermore, we have f (u,)-
Ur.
1l
f(ul) > Ei
by construction.
To charge Ei to the balls belonging to points in the path qj, we proceed as follows. Every point z of the interval [f(Ul), f(ur)] is contained in at least one ball
B(ui, cost(ui)). We will charge this point against the corresponding ball, in fact we
charge it to z E B(ui, cost(ui)). (It may be the case that many of the balls contain
z. We can just charge all of them.) Clearly this charges all of ei.
We are only left to see that every vertex is charged at most 2 c, · cost(v). But
this is easy; note that a point z E B(v, cost(v)) can only be charged in the round
corresponding to the segment of distance si in which z is contained. Secondly, notice
that in this round it can only be charged c times since v occurs in at most c, of the
paths ql,.. ., qj. This completes the proof of Lemma 9.5.
Combining Lemmas 9.4 and 9.5 finishes the proof of Lemma 9.3.
9.4.2
0
0
Line Embeddings and Distortion
Let (X, d) be a metric space. A map f : X -- R is called 1-Lipschitz if, for all
x,y E X,
If(x) -f (y)I
d(x, y)
Given a 1-Lipschitz map f and a demand function w X x X -t R+, we define its
average distortion under w by
avd,,(f) =
Ex(X,y
181
> y f(x ) - y)
f(x
We say that a weight function w is a product weight if it can be written as w(x, y) =
7r(x)ir(y) for all x, y E X, for some 7r: X -- R+.
I
We now state three theorems which
give line embeddings of small average distortion in various settings. The proofs of
these theorems are described in [90].
Theorem 9.6 (Bourgain, [43]). If (X, d) is an n-point metric space, then for every
weight function w : X x X -- R+, there exists an efficiently computable map f : X
R with avd,(f) = O(log n).
Theorem 9.7 (Rabinovich, [139]). If (X, d) is any metric space supported on a graph
which excludes a Kr-minor, then for every product weight wo : X x X -- R+, there
exists an efficiently computable map f
X --+R with avd,,o (f) = O(r 2 ).
Theorem 9.8 (Arora, Rao, Vazirani, [19]). If (X, d) is an n-point metric of negative
type, then for every product weight wo : X x X -
R+, there exists an efficiently
computablemap f X -- IR with avd, O(f) = O(;ogn).
We also recall the following classical result.
Lemma 9.9. Let (Y, d) be any metric space and X C Y. Given a 1-Lipschitz map
f :X
R, there exists a 1-Lipschitzextension f : Y
R, i.e. such that f (x) = f(x)
for all x E X.
Proof. One defines
f (y) = sup [f(x) - d(x, y)]
xEX
El
for all y E Y.
9.4.3
Analysis of the Vector Program
We now continue our analysis of the vector program from Section 9.3.3. Recall that
ir(i)(1 - x - y2) is the contribution of vertex i to the objective function. For every
i E V, define cost(i) = 4(1 - x - y). We will consider the metric space (V, d) given
by d(i,j) = (xi -
j) 2
(note that this is a metric space precisely because every valid
solution to the SDP must satisfy the triangle inequality constraints). The following
182
key proposition allows us to apply the techniques of Sections 9.4.1 and 9.4.2 to the
solution of the vector program.
Proposition 9.10. For every edge(i,j) E E, cost(i) + cost(j) > 2(xi - xj)2 .
Proof. Since (i, j) E E, we have xi yj = xj yi = 0, and recall that xi yi = xj yj = 0.
It followsthat
(x i - Xj)2 < 2[(xi + Yi - V)2 + (j + Yi- v) 2] < 2[(1-X2 _
(The first inequality followsfrom the fact that (xi-xj)
2
) + (1 -X2 _ i)
= ((xi+yi-v)-
(xj+y-v))2,
and from the fact that (x + y) 2 > 0, implying x 2 + y2 > 2xy and (x- y)2 < 2(x 2 + y 2 ).
Substitute x = xi + yi - v and y = xj + yi - v.)
Putting yj instead of yi in the above equation gives (xi - xj)2 < 2[(1 -2
_ y2) +
(1 - xj - y)]. Summing these two inequalities yields
2(x - xj)2
4[(1-
2 - y2) + (1 -2
Now, let U = supp(7r) = {i E V:
f : (U, d) -
7r(i)
_ yj2)]< cost(i) + cost(j).
0}, and put k = UI. Finally, let
R be any 1-Lipschitz map, and let f:
V -
R be the 1-Lipschitz
extension guaranteed by Lemma 9.9.
Then for any path vl, ... , v, in G, we have
m-1
If(vl)- f(m)
m
(xvi -xvi+)
< d(vl, Vm) < E d(vi, i+l) =
i=1
i=1
m
1 rn
(cost(vi) + cost(vi+l))
<
--2
i=1
m
< E cost(Vi).
i=1
where the third inequality is from Proposition 9.10. We conclude that f is pathcompatible with cost.
183
Defining a product demand by w(i, j) =
r(i)r(j) for every i, j
E V and capacities
ci = 7r(i), we now apply FINDCUT(G, f). If the best separator found has sparsity ao,
then by Lemma 9.3,
1
K 7(i)(1 -X
)
y2)
4
iEV
02
ci. cost(i) > 8K
4K
L
(i,j) 1f(i) -f(i)
8i,jEV
iEV
E (i,j)
8Koi,jEU
o
i,,EU('iJ)
- 4
If(i)-f()I
If(i)- f(i)I
EZi,jEu (i,j) d(i,j)
ao
4. avd(f)'
It follows that &cr(G)> ao/(4. avd, (f)). Since the metric (V, d) is of negative type
and w(., ) is a product weight, we can achieve avd,(f)
=
O(v/FgWk) using Theorem
9.8. Using this f, it followsthat FINDCUT(G,f) returns a separator (A, B, S) such
that c,(A, B, S) < O(x/J-ok) d&(G),completingthe analysis.
Theorem 9.11. Given a graph G = (V, E) and vertex weights 7r : V -- R+, there
exists a polynomial-time algorithm which computes a vertex separator (A, B, S) for
which
a,(A, B,S) < O( log) c,(G),
where k = Isupp(ir).
We extend this theorem to more general weights in Section 9.7.1. This is necessary
for some of the applications in Section 9.7.
9.5
Approximate Max-Flow/Min-Vertex-Cut Theorems
Let G = (V, E) be a graph with capacities {cv}vEV on vertices and a demand function
w : V x V --. R+. The maximum concurrent vertex flow of this instance is the
maximum constant e E [0,1] such that one can simultaneously route an e fraction of
184
each u-v demand w(u, v) without violating the capacity constraints.
Let Puv be the set of all u-v paths in G, and for each p E Puv, let pUVdenote the
amount of the u-v commodity that is sent from u to v along p. We now write a linear
program that computes the maximum concurrent vertex flow.
maximize
6
S
subject to
puv>E6 .(u,
v)
u,v E V
PEPuv
E E pUV,,
wEV
u,vEV pEPuv:wEp
puV > 0
u, v E V, p E Pu,,
We now write the dual of this LP with variables
{SV}VEV
and {tUv}U,veV.
5 CvSv
minimize
vEV
subject to
E sw >ev
p Euv,Vu,
vEV
wEp
E w(uv)euv > 1
u,vEV
eV >o, Sv o0
u, vE V.
Finally, define
dist(u, v) = min
pE~
'uv
: Sw.
wEp
By setting u = dist(u, v), we see that the above dual LP is equivalent to the following.
minimize
5CvSv
vEV
subject to
(u, v) dist(u, v) > 1.
u,v
Remark. The dual LP above is derived by defining the length of a path p = l,,
185
,
k
to be
El<i<k
Svi. However, a "cleaner" definition would take only half of the con-
tribution of the endpoints of the path, giving it length s 1 /2 +
E2<i<k-1
Svi + Svk/2.
Under this second definition, each edge (u, v) has length (su + sv)/2, and the length
of a path is the sum of the lengths of its respective edges. In terms of approximating
vertex cuts, the two definitions differ by a factor of at most two, which makes little
difference for us. We choose to work with the first definition because it simplifies some
of the notation. However, in Section 9.5.3 we shall switch to the second definition.
9.5.1 Rounding to Vertex Separators
Observe that any vertex separator (A, B, S) yields an upper bound on the maximum
concurrent flow in G. The upper bound is of the form
acapw(A, B
cap(S),
)=
uEAUS EVEBUSw(U, V)
The numerator is the capacity of the separator, while the denominator is the amount
of demand that must be sent through it. To see how tight this upper bound is in
general, we can take the dual of the max-concurrent-flow LP from the previous section
and round it to a vertex separator while increasing the cost by at most some factor.
We will write a = CaP,, if the capacity and demands are clear from context. We
note that the dual LP is a relaxation of a(G), since every vertex separator (A, B, S)
gives a feasible solution, where s, = 1/A if v E S and s, = 0 otherwise. In this
case dist(u,v) > 1/A if u E A U S and v E B U S or visa-versa, so that setting
A=
ZuEAUS,EBUS
W(U, v) yields a feasible solution.
9.5.2 The Rounding
Before presenting our approach for rounding the LP, let us recall a typical rounding
approach for the case of edge-capacitated flows. In the edge context [131, 21], one
observes that the dual LP is essentially integral when dist(-, ) forms an L 1 metric.
To round in the case when dist(-, ) does not form an L 1 metric, one uses Bourgain's
theorem [43] to embed (V, dist) into L1 (with O(log n) distortion, that translates to
186
a similar loss in the approximation ratio), and then rounds the resulting L1 metric
(where rounding the L 1 metric does not incur a loss in the approximation ratio). This
approach is not as effective in the case of vertex separators, because rounding an L1
metric does incur a loss in the approximation ratio (as the example below shows), and
hence there is not much point in embedding (V, dist) into L1 and paying the distortion
factor.
The discrete cube. Let G = (V, E) be the d-dimensional discrete hypercube {0, 1}d.
We put c, = 1 for every v E V, and w(u,v)
well-known that accaP,'(G) =
(1/(2d.V))
= 1 for every pair u,v
E V.
It is
[111]. On the other hand, consider the
fractional separator (i.e. dual solution) given by s = 10 · -.
Note that dist(u, v) is
proportional to the shortest-path metric on the standard cube, hence Euv dist(u, v) >
1, yielding a feasible solution which is a factor O(i)
away from a(G).
It follows that even when (V, dist) is an L1 metric, the integrality gap of the dual
LP might be as large as
(/logn).
Rounding with line embeddings.
The rounding is done as follows. Let {sV}VEv
be an optimal solution to the dual LP, and let dist(., ) be the corresponding metric
on V. Suppose that the demand function w: V x V -R+
is supported on a set S,
i.e. w(u, v) > 0 only if u, v E S, and that ISI = k. Let f: (S, dist)
-*
R be the map
guaranteed by Theorem 9.6 with avd,(f) = O(log k), and let f: (V, dist) -1 R be the
1-Lipschitz extension from Lemma 9.9.
For v E V, define cost(v) = s. Then since f is 1-Lipschitz, for a path v 1, v2 , ... , Vm
in G, we have
m
i=l
cost(v) > dist(vl, vm) > If(vl)-
f(Vm) ,
hence f is path-compatible with cost.
We now apply FINDCUT(G, f). If the best separator found has sparsity
187
o, then
by Lemma 9.3,
Ecvsv
v
E cv. cost(v) > oaoE w(u, v)lf(u)-
f(v)I
u,vEV
v
=
ao E w(u,v) If(u)- f(v)I
u,vES
w(u, v) dist(u, v) > Q
> Q logk
ogk
u,vEV
Theorem 9.12. For an arbitraryvertex-capacitatedflow instance, where the demand
is supported on a set of size k, there is an O(log k)-approximate max-flow/min-vertexcut theorem. In particular, this holds if there are only k commodities.
9.5.3
Excluded Minor Families
Here we shall switch to the "cleaner" definition for path lengths that is presented
in the remark following the description of the dual LP. This allows us to view path
lengths as being induced by edge lengths. A consequence of this is that if the graph
G excludes some fixed graph H as a minor, then the metric arising from the LP dual
is an H-excluded metric. It follows that applying Theorem 9.7, yields a better result
when G excludes a minor and when we have a product demand function w(u, v).
Theorem 9.13. When G is an H-minor-free graph, there is an O(IV(H)1 2 )-approximate
max-flow/min-vertex-cut theorem with product demands. Additionally, there exists an
O(IV(H)12 ) approximation algorithm for finding min-quotient vertex cuts in G.
9.6
An Integrality Gap for the Vector Program
Consider the hypercube graph. Namely, the n vertices of the graph (where n is a power
of 2) can be viewed as all vectors in {+1}l
° gn ,
and edges connect two vertices that
differ in exactly one coordinate. Every vertex separator (A, B, S) has a(A, B, S) >
1/O(n
log/n). This follows from standard vertex isoperimetry on the cube [111]. We
show a solution to the vector program with value of O(n/ log n), proving an integrality
188
ratio of Q(l-og r) for the vector program, and implying that our rounding technique
achieves the best possible approximation ratio (relative to the vector program), up
to constant multiplicative factors.
In the solution to the vector program, we describe for every vertex i the associated
vectors xi and yi. The vectors si will not be described explicitly, but are implicit,
using the relation si
v - xi - Yi.
Each vector will be described as a vector in
1 + n log n + 2(n - 1) dimensions (even though n dimensions certainly suffice). Our
redundant representation in terms of number of dimensions helps clarify the structure
of the solution.
To describe the vector solution, we introduce two parameters, a and b. Their
exact value will be determined later, and will turn out to be a = 1/2 - E(1/log n),
b = O(1/n
logn). We partition the coordinates into three groups of coordinates.
G1. Group 1, containing one coordinate. This coordinate corresponds to the direction of vector v (which has value 1 in this coordinate and 0 elsewhere). All xi
and yi vectors have value a on this coordinate.
G2. Group 2, containing n identical blocks of logn coordinates. The coordinates
within a block exactly correspond to the structure of the hypercube. Within
a block, each xi is a vector in {b}l
°g n
derived by scaling the hypercube label
° g n) by a factor of b. Vector yi is the
l
of vertex i (which is a vector in {+1}
negation of vector xi on the coordinates of Group 2.
G3. Group 3, containing 2 identical blocks of n - 1 coordinates. The coordinates
within a block arrange all the xi vectors as vertices of a simplex. This is done
in the following way. Let Hn be the n by n Hadamard matrix with entries +1,
obtained by taking the (log n)-fold tensor product of the 2 by 2 the matrix H2
that has rows (1, 1) and (1, -1). The inner product of any two rows of Hn is 0,
the first column is all 1, and the sum of entries in any other column is 0. Remove
the first column to obtain the matrix Hn. Within a block, let vector xi be the
ith row of H, scaled by a factor of b. Hence within a block, ix i = b2 (n - 1),
189
and xixj = -b 2 for i f j. Vector yi is identical to xi on the coordinates of
Group 3.
We now show that the triangle constraints are satisfied by our vector solution.
Recall (see Section 9.3) that there is some flexibility in the choice of which triangle
constraints to include in the vector program (and likewise for many other constraints
that are valid for 0/1 solutions but are not used in our analysis). We shall address
here a subset of the triangle constraints that is larger than that actually used in the
analysis of our rounding algorithm.
There are five sets of vectors from which we can take the three vectors that
participate in a triangle constraint: X (the xi vectors), Y (the yi vectors), S (the
si vectors), v and 0. In our analysis we used only triangle constraints over vectors
from X. Here we show that all the triangle constraints that involve only vectors from
X U Y are satisfied. All vectors in X U Y have the identical value a in their first
coordinate, and in every other coordinate they take only values from ±tb. Hence per
coordinate, every quadratic constraint that holds for all ±1 vectors (including, but
not limited to, the triangle constraints) is satisfied for all xi and yi vectors.
We let K = Ei,jEv(xi - xj)2 = e(n3 b2 log n). The value of the parameters a and
b is governed by the following three constraints.
1. The exclusion constraints imply that
a2 - nb2 logn + 2b2 (n- 1) = O
2. The edge constraints (and the fact that edges connect vertices of Hamming
distance 1) imply that
a2 - nb2 (logn- 2) - 2b2 = 0
3. The sphere constraints imply that
a = a2 + nb2 logn + 2b2(n - 1)
190
Hence we have a system of three equalities in two unknowns (a and b). This
system is consistent, because the first two equalities are in fact identical (due to our
careful choice of number of blocks in each group). They both give:
a2 + (-n log n + 2n- 2)b2 = 0
By setting b = a/
n log n- 2n
2 the first two equalities are satisfied. The third
equality now reads a = a2 (2 + e) for some E = O(1/ log n). This equality is satisfied
by taking a roughly equal to 1/2 - E/4, which is 1/2 - E(1/ log n).
It follows that in the vector solution all s = 1-
- yi is 0(1/log n) for every
i E V. Hence our vector solution has value
1
1
2
K i Zi
E(n log n)
Finally we note that rather than have only one coordinate in Group 1, we can
have (a/b) 2 = n log n - 2n + 2 coordinates, and give the x and y vectors values b in
these coordinates. Then all x and y vectors become vertices of a 2n log n-dimensional
hypercube (of side length b). We see that even in this special case, the integrality gap
remains Q(VIo-g).
9.7 Balanced Vertex Separators and Applications
9.7.1
More General Weights
An important generalization of the min-ratio vertex cut introduced in Section 9.3 is
when a pair of weight functions rl, r2 : V - IR+ is given and one wants to find the
vertex separator (A, B, S) which minimizes
rw12(A, B, S)
We can also give an O(/-oi
_T2(A U S) . 7 (B U S)'
2
) approximation in this case, where again k = Isupp(ir2 ).
191
Let
&,rl,wn
2(A, B, S) = ri(S)/[7r2(A)
where r2(A) > 7r2 (B). Also define c°,,
2 (G)
relaxation for
,
2 (G).
and &,
K Eie
changing the vector program to minimize
7r2 (B U S)],
2(G)
as before. Note that by
7rl(i)(1 - x? - y2), it becomes a
Similarly, the rounding analysis goes through unchanged to
yield a separator (A, B, S) with
Cr 1,,,2 (A, B,
S) < O( l/ogk) &o,,1 2 (G).
The only difficulty is that if (A*, B*, S*) is the optimal separator, the relation
between the two notions,
,,,,, 2 (A*, B*, S*) and
,r,, 2(A*, B*, S*), is no longer as
clear. If they are not within a factor of 2, then it must hold that
r2 (A* U S*) >
27r2 (A*), i.e. r 2 (S*) > 7r2(A*). (Where we assume that 7r2 (A*) > 7r2 (B*).)
In this case,
r2(S*)2 < 4
(G).
Hence it suffices to find an approximation for a different problem, that of finding a
subset S C V which minimizes the ratio rl(S)/ir 2 (S) 2 . This problem can be solved
in polynomial time (see e.g. [90]).
Theorem 9.14. Given a graph G = (V, E) and vertex weights 7r, r2
V -, R+, there
exists a polynomial-time algorithm which computes a vertex separator (A, B, S) for
which
a l 72(A, B, S) •
( logk) a-,,- 2 (G),
where k = Isupp(r 2)l.
We also note that Theorem 9.14 with general weights rt, i 2 is useful for certain
hypergraph partitioning problems [129].
192
9.7.2
Reduction from Min-Ratio Cuts to Balanced Separators
In this section, we sketch a pseudo-approximation for finding balanced vertex sepaE (0, 1),
rators in a graph G = (V, E). Let W C V be an arbitrary subset of V. For
we say that a subset X C V is a 6-vertex-separator (with respect to W) if every connected component C of G[V \ X] has IC n W I <
IWl. Our
goal in this section is to
show that we can find a 3-vertex-separator X C V whose size is within an O(O) factor
of the optimal
-vertex-separator of G, whenever we can find approximate min-ratio
cuts in G within factor /3. This technique is standard (see [129]).
Let m = IWI, and for any subset U C V, define UIw = IU n WI.
The algorithm.
Let 7rl(v) = 1 for every v E V, and 7r2 (v) = 1 if v E W and 7r2 (v) = 0 otherwise.
These are the weights for the numerator and denominator, respectively, i.e. we assume
that we have a P-approximation for a,,
7
2(-). We
maintain a vertex separator S C V.
Initially, S = 0. As long as there exists some connected component U C V in G[V\S]
with UIw > 2W[,
we use our/3-approximation to find a minimum-ratio vertex cut
S' in G[U] which is within : of optimal. We then set S
-
S U S' and continue.
The analysis. Let S be the final vertex separator. By construction, it is a -vertex
separator since every connected component U of G[V \ S] has IUIw < IW.
Let
T C V be an optimal -vertex separator.
Claim 9.15. ISI < O(/)ITI.
Proof. We know that T separates G into two pieces, call them AT, BT C V such that
IAT U TIw, IBT U TIw > 1lWI. Suppose we are at a step where IU w >
2
Wl. Let
(A', B', S') be the vertex separator in G[U] that we find by running our min-quotient
cut algorithm with ratio 3, and suppose that IA'Iw > IB'Iw. We know that
IS'l
IA' Sw
<TI
B'
S-
12TI
I(AT UT) n Ulw. (BT U T) n UIw 193
m2
where the final inequality follows because IUlw > 2.
It follows that
120IT(IB'lw + IS'Il)
To see that ISI < O(3)ITI, it suffices to see that when we sum IB'I, + IS'I, over all
iterations, the value is at most O(m). But since we throw away the vertices of B' US'
in every iteration (and recurse only on A'), the sum is clearly at most m.
[O
9.7.3
Approximating Treewidth
The notion of treewidth has numerous practical applications (see e.g. [33]) and thus a
large amount of effort has been put into determining treewidth, which is NP-complete
even when the input graph is severely restricted (see the discussion in Section 2.4 for
a brief history).
From the approximation viewpoint, Bodlaender et al. [34]gave an O(log n) approximation algorithm for treewidth on general graphs. Amir [11] improved the approximation factor to O(log opt) where opt is the actual treewidth of the graph. Constantfactor approximations for treewidth were obtained on AT-free graphs [42, 41] and
on planar graphs [155]. The approximation for planar graphs is a consequence of
the polynomial-time algorithm given by [155] for computing the parameter branchwidth, whose value approximates treewidth within a factor of 1.5. Recently, [12]
obtained a new approximation algorithm for treewidth in planar graphs with a constant factor slightly worse than 1.5, and the authors of [72] (see also Chapter 2)
derived a polynomial-time algorithm for approximating treewidth within a factor of
1.5 for single-crossing-minor-free graphs, generalizations of planar graphs. A wellknown open problem is whether treewidth can be approximated within a constant
factor. We show that this is indeed the case for every family of graphs that excludes
H as a minor, for an arbitrary fixed graph H. For general graphs we improve the
approximation factor to 0(logopit)
where opt is the actual treewidth of the graph.
These improvements have several implications, including better approximation
194
algorithms for several other problems like pathwidth, minimum front size, and minimum height elimination tree. They also improve the running time of approximation
schemes and fixed-parameter algorithms for several NP-complete problems on graphs
of bounded genus, or more generally, graphs excluding a fixed graph as a minor.
Now we are ready to state our approximation result for treewidth.
Theorem 9.16. There exists a polynomial time algorithm that find a tree decomposition of width at most O(V/logtw(G) tw(G)) for a generalgraphG and at most
O(IV(H)12 tw(G)) for an H-minor-free graph G.
Proof. The proof is a straightforward modification of the proof of Amir [11]. One
recursively uses a balanced vertex-cut algorithm and then proves the aforementioned
width bound by induction. In [11] one uses a polynomial-time algorithm that, given
a graph G = (V, E) and a set W C V, finds a 3-vertex separator S C V of W in G
of size O(log WI k), where k is the minimum size of a 1-vertex separator of W in G.
Now using Theorems 9.14 and 9.13 and the results of Subsection 9.7.2, we can replace
O(log WI k) with O(v/log WI k) for general graphs and O(r 2 k) for Kr-minor-free
graphs, and thus we obtain the desired result.
[1
By using the result of [34] instead of [11] in the proof of Theorem 9.16, we can
obtain a tree decomposition of "logarithmic depth" (in terms of IV(G) ) with width
at most O(V/ 1gn tw(G)) for a general graph G and at most O(IV(H)
2
tw(G)) for
an H-minor-free graph G.
Improving the approximation factor of treewidth improves the approximation factor for several other problems as follows.
Corollary 9.17. There exist O(Vlogopt) (resp., O(IV(H)1 2 )) approximation algo-
rithms for branchwidth, minimum front size and minimum size of a clique in a
chordal supergraph of a general (resp., H-minor-free) graph G. Additionally, there
are O(/logoptlogn)
(resp., O(IV(H)12 logn)) approximation algorithmsfor path-
width, minimum height elimination ordertree, and search numberin a general(resp.,
H-minor-free) graph G.
195
The reader is referred to Bodlaender et al. [34] and Leighton and Rao [129]to see
the corresponding exact definitions, references, and the proofs of Corollary 9.17 (the
proofs follow almost identical to those of Theorems 17 and 18 of [34] and the fact that
treewidth and branchwidth are within a factor 1.5 of each other (see Theorem 2.15).
Improving the approximation factor for treewidth has a direct improvement on
the running time of approximation schemes and subexponential fixed-parameter algorithms for several NP-hard problems on graphs families which exclude a fixed minor.
In such algorithms finding the tree decomposition of almost minimum width, on which
we can run dynamic programming, plays a very important role. More precisely, as
mentioned in Chapter 8, Demaine and Hajiaghayi [70, 71] show how one can obtain
PTASs for almost all bidimensional parameters on planar graphs, single-crossingminor-free graphs and bounded-genus graphs. In fact, their approach can be extended
to work on apex-minor-free graphs for contraction-bidimensional parameters and on
H-minor-free graphs, where H is a fixed graph, for minor-bidimensional parameters.
However currently they obtained quasi-polynomial-time approximation schemes for
these general settings. The only barrier to obtain PTASs for these general settings is
obtaining a constant-factor polynomial-time approximation algorithm for treewidth
of an H-minor-free graph for a fixed H (this is posed as an open problem in [70]).
Using Theorem 9.16, we overcome this barrier and obtain PTASs for contractionbidimensional parameters in apex-minor-free graphs and for minor-bidimensional parameters in H-minor-free graphs for a fixed H. As an immediate consequence, we
obtain the following theorem (see [70, 71] for the exact definitions of the problems
mentioned below).
Theorem 9.18. There are PTASs for feedback vertex set, vertex cover, minimum
maximal matching, and a series of vertex-removal problems in H-minor-free graphs
for a fixed H. Also, there are PTASs for dominating set, edge dominating set, rdominating set, connected dominating set, connected edge dominating set, connected
r-dominating set, and clique-transversalset in apex-minor-freegraphs.
Among the problems mentioned above, PTASs for vertex cover and dominating
196
set (but not its other variants) using a different approach were known before (see
e.g. [103]).
197
198
Chapter 10
Open Problems Regarding
Bidimensionality
In this thesis, we introduced the theory of bidimensionality and its applications in
algorithmic graph theory. However, still several combinatorial and algorithmic open
problems remain in the theory of bidimensionality and related concepts. These problems are usually more general than those mentioned in the end of some previous
chapters.
One interesting direction is to generalize bidimensionality to handle general graphs,
not just H-minor-free graph classes. As mentioned in Section 1.5, the natural generalization of minor-bidimensionality still yields a parameter-treewidth bound, but it is
very large. This direction essentially asks for the size of the largest grid minor guaranteed to exist in any graph of treewidth w. Robertson, Seymour, and Thomas [151]
proved that every graph of treewidth larger than
5
2 0 2r
has an r x r grid as a minor,
but that some graphs of treewidth Q(r2 lg r) have no grid larger than O(r) x O(r),
conjecturing that the right requirement on treewidth for an r x r grid is closer to
(r2 lgr) lower bound.
If this conjecture is correct, we would obtain nearly
as good parameter-treewidth
bounds for minor-bidimensional parameters as in the
the
H-minor-free case. A similar generalization of parameter-treewidth bounds beyond
apex-minor-free graphs is not possible for all contraction-bidimensional parameters,
e.g., dominating set [62],but it would still be quite interesting to explore an analogous
199
"theory of graph contractions" paralleling the Graph Minor Theory. Such a theory
would be an interesting and powerful tool for handling problems that are closed under
contractions but not minors, and therefore deserves more focus.
Another interesting direction is to obtain the best constant factors in terms of the
fixed excluded minor H. These constants are particularly important in the context
of the exponent in the running time of a fixed-parameter algorithm. At the heart of
all such constant factors is the lead constant in Theorem 1.6. This factor must be
Q(v IV(H)lg I[V(H)I), because otherwise such a bound would contradict the lower
bound for general graphs. An upper bound near this lower bound (in particular,
polynomial in IV(H) ) is not out of the question: the bound on the size of separators
in [9] has a lead factor of IV(H)13 /2 . In fact, Alon, Seymour, and Thomas [9] suspect
that the correct factor for separators is
(IV(H)I), which holds e.g. in bounded-genus
graphs. We also suspect that the same bound holds for the factor in Theorem 1.6,
which would imply the corresponding bound for separators.
A third interesting direction is to generalize the polynomial-time approximation
schemes that come out of bidimensionality to more general algorithmic problems that
do not correspond directly to bidimensional parameters. One general family of such
problems arises when adding weights to vertices and/or edges, and the goal is e.g. to
find the minimum-weight dominating set. It is difficult to define bidimensionality of
the corresponding weighted parameter because its value is no longer well-defined on
an r x r grid: the parameter value now depends on the weights of vertices in such
a grid. Another family of such problems arises when placing constraints (e.g., on
coverage or domination) only on subsets of vertices and/or edges. Examples of such
problems include Steiner tree [17] and subset feedback vertex set [89]. Again it is
difficult to define bidimensionality in such cases because the value of the parameter
on a grid depends on which vertices and/or edges of the grid are in the subset.
200
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